Infinity minus an infinity yields logically impossible scenarios. Notably, one can take away identical quantities from identical quantities and arrive at contradictory remainders.

On the face of it, this claim appeals to our intuitive understanding of subtraction. If I were to claim that there exists some Integer, , such that and , then we stumble upon the contradiction that . Subtracting identical quantities from identical quantities should yield identical results.

Belief Map further cites the work of William Lane Craig and James Sinclair for a particular example of how infinities can yield such contradictions.

For example, if we subtract all the even numbers from all the natural numbers, we get an infinity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the identical number of numbers and yet did not arrive at an identical result. In fact, one can subtract equal quantities from equal quantities and get any quantity between zero and infinity as the remainder. For this reason, subtraction and division of infinite quantities are simply prohibited in transfinite arithmetic – a mere stipulation which has no force in the nonmathematical realm.

Unfortunately, this example is not quite so logical as its authors would like it to be. The reason for this is that they don’t seem to have a very good grasp of the mathematics which they are trying to discuss. The final statement of this paragraph makes a blatantly false claim: “For this reason, subtraction and division of infinite quantities are simply prohibited in transfinite arithmetic.” It is not the case that subtraction and division are “simply prohibited in transfinite arithmetic.” Rather, subtraction and division are not defined in these cases. This is a very important distinction.

Imagine, for a moment, that I have asked you to perform the calculation “seven vike four.” You would rightly ask me what ‘vike’ means? How does one ‘vike’ two numbers? Until there is a definition for viking numbers, it is not possible for anyone to vike those numbers. It is not that we are prohibited from viking numbers. It is simply that the phrase is undefined and, therefore, meaningless.

Similarly, let’s say that I asked you to perform the calculation “seven divided by zero.” Now, despite the fact that division is well-defined for almost all numbers, you may recall from your grammar school math classes that division is not defined when dividing by zero. In this case, once again, we are not prohibited from dividing by zero. Dividing by zero is simply not a meaningful concept.

Similarly, in the field of mathematics known as “transfinite arithmetic,” subtraction and division are not defined operations. Despite the similar name, “transfinite arithmetic” is not the same arithmetic which children learn in grammar school. It is a collection of rules for manipulating sets. It doesn’t matter that these operations are well-defined for other areas of mathematics– they are not defined for transfinite arithmetic. Again, these operations are not prohibited, as Dr. Craig claims; they are simply meaningless.

But doesn’t Dr. Craig’s example cite clear ways of subtracting from infinite sets? How can it be undefined if there exist ways to do it?

We are not saying that one cannot choose and eliminate any arbitrary number of elements from a given infinite set. Rather, what we are saying is that there is no general definition which can account for all cases of subtraction between two different sets in transfinite arithmetic. The manner in which one eliminates the elements from an infinite set can change the result, and therefore there is no single, correct way to do it. So, contrary to Dr. Craig’s claim, it is not the case that we are prohibited from removing elements of an infinite set, in mathematics; we simply do not have a general operation for subtracting any given set from any other given set, in transfinite arithmetic. Special cases can always be considered by themselves.

When different special cases of performing an operation can be done in order to produce different results, we say that the more general statement is *indeterminate* in that case. For example, as we mentioned earlier, division by zero is not defined in the arithmetic which we all learned in grammar school. However, when mathematicians see the statement they immediately recognize it as an indeterminate form. Depending on the special case, the mathematical phrase can be equal to any number.

Take, for instance, the phrases and . In both cases, these limits resolve to . However, a simple application of L’Hopital’s rule tells us that while . I can just as easily show cases where is equal to 5 or to 17 or to π or to any other number. This is not inconsistent or contradictory, because is not a number; it is a mathematical phrase.

Similarly, we can imagine two numbers and which are both infinite. The phrase is indeterminate. Depending on the particular case, could result in any number. This is not a contradiction. This is not an inconsistency. This is simply an indeterminancy.

The other examples which apologists usually bring out in support of their opposition to actual infinites tend to fall to similar problems. They are misunderstandings or misstatements of the underlying mathematics. The simple fact of the matter is that the idea that actual infinities are logically and metaphysically possible has not even been controversial in the Philosophy of Mathematics for a century. As such, when an apologist makes pronouncements upon the nature of mathematics, we should be more than a little skeptical.

]]>However, today, I’m going to break that trend. I was recently directed to an article written in November of 2017 entitled Nine Points about Biblical Slavery and Skeptics’ Condemnation of the Bible. The author of the article intends the piece to be a defense of the Bible against complaints regarding the manner in which its constituent documents treat the institution of slavery. The topic is an understandably sore one for a great number of people. Opponents of religion are quite often quick to point out that the Ten Commandments, a list of some of the supposed most important tenets of morality, includes things like, “Keep the Sabbath holy” and “Honor thy father and mother,” but omits something like, “Thou shalt not own another person as your property.” In fact, the Bible never offers any clear proscription against slavery, and indeed seems to sanction the practice in a number of places. Given the incredibly strong modern moral aversion to the idea of slavery, Christian apologists have attempted to take up the task of showing that these modern detractors hold deep misunderstandings of what the Bible actually says about slavery.

The reason I’m breaking my usual avoidance of topics of morality to discuss this one is that I honestly don’t care about the metaphysical foundations of our ethics, in regards to slavery. All I care about, regarding this topic, are the answers to two, simple questions.

Is it morally acceptable to own another human being as property? Does the Bible sanction the owning of another human being as property?

Even before the author gets into any of the titular “Nine Points,” a rather curious statement is made.

The horrific treatment of slaves in the United States between 1619 and 1865 has led many Bible skeptics to question the often-mentioned practice of Biblical slavery, with the assumption that the systems of the U.S. were similar to those in Biblical times.

Even if there are skeptics who assume that “the systems of the U.S. were similar to those in Biblical times,” focusing on the American institution of slavery is a complete Red Herring. Is it morally acceptable to own another human being as property? Does the Bible sanction the owning of another human being as property?

The Bible is an historical textbook, which documented actual events in our history.If we wanted a book devoid of actual events and our sometimes ugly history, the Bible would have been a fairy tale instead of an historical text.

While I would disagree with the flat, categorical assertion that “the Bible is an historical textbook,” I will agree that there are certainly texts within that anthology which are intended to convey historical events. The article is absolutely correct to note that a discussion of history which glosses over the negative parts would be far less trustworthy. However, it is quite curious, indeed, that the article seems to be admitting that the discussion of slavery in the Bible amounts to a discussion of “our sometimes ugly history.”

Is it morally acceptable to own another human being as property? Does the Bible sanction the owning of another human being as property?

Slavery was an integral part of the functioning of societies in Biblical times.

- “As much as two thirds of the Roman empire were slaves (before the first century it was as high as 90%). By the first century AD an increasingly large number of slaves were being freed—so much so that Caesar had to write up laws that governed the procedure!” (Wallace, 2004).
- Ancient people would have had a difficult, if not impossible time of building societies without the manpower of slaves.

It seems that our Christian Apologist feels the need to be an apologist for slavery in order to defend the faith. Yes, a great deal of ancient society was built upon the backs of slaves. One could certainly build a strong argument that the cultural progression we now know from history would have been severely retarded without the institution of slavery. Is it the author’s intention to imply that this fact therefore implies that slavery is ethically vindicated?

Is it morally acceptable to own another human being as property? Does the Bible sanction the owning of another human being as property?

Slaves were captives of war, children of slaves, abandoned children, criminals, or debt-servants. Slavery was not due to skin color.

Does the author intend to imply that only slavery based on race is immoral? Are we to believe that enslaving captives of war, children of slaves, abandoned children, criminals, and debt-servants is completely ethical behavior? Is it the author’s belief that slavery is justifiable, but racism is abhorrent?

4. Three provisions for slaves in Israel differ markedly from the rights afforded them in the New World (see above source for additional details).

Anti-harm laws: When an employer accidentally harmed a slave, the slave was to go free, as per Exodus 21: 26-27. Further, in the same passage, if an owner put a slave to death, the owner was to be put to death. In the New World, slaves did not have such rights and owners were not punished for harming slaves.Anti-kidnapping laws: Kidnapping was condemned and punishable by death, as per Exodus 21:16 and Deuteronomy 24:7. In the New World, slavery began by the kidnapping of people in Africa.Anti-return laws: Deuteronomy 23: 15-16 states that Israel is to offer a safe haven to harbor runaway slaves, which strongly contrasts the “Fugitive Slave Law” of the United States.

The article’s treatment of what it classifies as “anti-harm laws” is extremely misleading. The passage in question doesn’t make reference to “an employer” or “accidental harm,” at all. Rather, the passage explicitly refers to a slaveowner deliberately striking a slave in such a way as to cause that slave to lose an eye or a tooth. In such an instance, Exodus calls for the slave to be freed. However, there is absolutely no proscription against the slaveowner injuring the slave in other ways, whether intentionally or accidentally. Further, the article completely mischaracterizes the punishment for a slaveowner killing a slave. I’ll let the actual passage speak for itself, here:

“*When a slaveowner strikes a male or female slave with a rod and the slave dies immediately, the owner shall be punished. But if the slave survives a day or two, there is no punishment; for the slave is the owner’s property.*” **–Exodus 21:20-21 (NRSV)**

The discussion of anti-kidnapping laws are completely beside the point. Whether a person enters into slavery by having been kidnapped or by some other means, that person still becomes the property of the slaveowner.

Yes, Deuteronomy 23:15-16 states that slaves who escape foreign countries are to be allowed to reside in Hebrew lands and are not to be oppressed. This does not, in any way, require Hebrew slaves to be freed, nor prevent the Hebrews from acquiring new slaves, nor discourage the Hebrew institution of slavery in any way.

On the contrary, Exodus and Deuteronomy explicitly sanction the purchase and acquisition of slaves. Exodus 21:2-6 and Deuteronomy 15:12-18 set down the rules by which slaves are to be acquired if they are Hebrews. Leviticus 25:44-46 give provisions for acquiring foreign slaves, who can be willed to one’s children as inherited property and are slaves for their whole lives.

5.

We should consider the New Covenant of the New Testament when discussing Biblical Slavery.

The article’s implications, here, seem to be that the New Covenant causes God’s moral prescriptions to change. I don’t have any problem with this, in and of itself, but it does seem to contradict the usual Christian position that morality is objective and immutable.

6.

The New Testament treatment of slaves demonstrated kindness

- In the book of Philemon (15-21), we see that Philemon owned a slave – and the fine treatment Paul afforded the slave

- “Some critics claim, “Jesus never said anything about the wrongness of slavery.” Not so. He explicitly
opposedevery form of oppression in His mission ‘to proclaim release to the captives … to set free those who are oppressed’ (Luke 4:18 NASB^{1}; cp. Isaiah 61:1)” (Copans, 2011).- “The New Testament presupposes a fundamental equality because all humans are created in God’s image (James 3:9). Yet, an even deeper unity in Christ transcends human boundaries and social structures: no Jew or Greek, slave or free, no male and female, as all believers are all ‘one in Christ Jesus’ (Galatians 3:28; cp. Colossians 3:11)” (Copans, 2011).
- Unlike Christians, Romans believed slaves were a “living tool, just as a tool is an inanimate slave. Therefore, there can be no friendship with a slave” (Nichomachean ethics 8.11).
- Christians freed slaves with estimates of at least 15,600 between 95 and 400 A.D. (Lecky, 1927).
- With the emergence of Christianity came the belief that Christians wouldn’t enslave other Christians (Rutgers, n.d.).

The book of Philemon is quite an interesting one, as regards slavery. It is a very short letter written by Paul to a fellow Christian named Philemon. In the epistle, Paul implies that Philemon owned a slave named Onesimus and that the slaveowner had given Onesimus in service to Paul since Philemon had no use for the man. Paul is returning the slave, Onesimus, to his owner, Philemon, accompanied by this letter. Paul tells Philemon that he has become very close to Onesimus– so close, in fact, that he has adopted Onesimus as his son. He implores– and even threatens to command– Philemon to treat Onesimus no longer as a slave, but rather as a brother.

While this book is often cited by apologists as evidence that slavery was opposed in the New Testament, the simple fact of the matter is that it does not evidence this, at all. Paul does not ask Philemon to free *all* of his slaves; he only cares about a single, particular slave. He does not implore that Onesimus be freed because slavery is immoral; he asks that Onesimus is freed because he has become extremely close with the man. Paul is asking Philemon to free Onesimus, but he sends the slave back to his owner still a slave and with no guarantee that Philemon will actually grant freedom.

The article then moves to referencing Luke 4:18 as an example of Jesus condemning slavery. However, a simple reading of the passage in its actual context disconfirms that eisegetical interpretation of the text. Neither Jesus nor the passage from Isaiah which Jesus quotes are discussing the notions of slavery or oppression or captivity as a whole. They are both referring directly to Israel and God’s chosen people. The Lukan passage is intended to show that Jesus is claiming the mantle of the Messiah, the anointed one of God sent to redeem God’s people. It is certainly not a flat proscription against slavery.

On the contrary, Jesus never offers a single objection to slavery, as an institution, but rather discusses it as a perfectly normal and acceptable occurrence. A number of Jesus’ parables make reference to slaves and their masters, and never in any way which might imply that the practice of slavery is immoral. According to the author of Luke, Jesus proclaims, “Blessed are those slaves whom the master finds alert when he comes” (12:37, NRSV). Other New Testament authors are even more forceful in their acceptance of slavery. Both Ephesians 6:5 and 1 Peter 2:18 exhort slaves to obey their masters, and explicitly denote that they should be glad to do so because it is the morally correct thing to do.

While some New Testament authors most certainly did claim that slaves and free-men were spiritual equals, this does not imply that they are societal equals or that the practice of owning another human being as property is, in any way, immoral.

The article’s remaining claims on this subject– that Roman law viewed slaves in a worse light than Christians, that Christians freed 15,600 slaves over the course of three centuries, and that Christians adopted a policy of not enslaving other Christians– are all Red Herrings. None of these things are related in the Bible, itself; and even if they were, none of these things are a condemnation of the practice of slavery, as a whole.

7.

Judging God’s actions to permit slavery or to terminate a particular society is faulty when we lack God’s omniscience and presence in the past, present, or future.

God’s omniscience and omnipresence are entirely irrelevant.

8.

Judging the practice of slavery in the Old Testament or New Testament requires assessing it against an objective moral standard, not a relative (society-specific, fluctuating) standard.

Judgment does not require an objective moral standard. Judgment can most certainly be passed on a relative moral standard, and is done so all the time. However, that is entirely irrelevant. Right now, I’m not attempting to judge the practice of slavery, at all. I’m asking two very simple questions regarding one’s view on the topic.

9.

Before judging societies for what they were during Biblical times, which consisted of high slavery populations and harsh conditions, consider the true message of the Bible and its entire purpose, which was to pave the path for Jesus.

Is the implication, here, that slavery is perfectly ethically sound so long as it is a tool for paving the path to Jesus? Are we to believe that the practice of slavery was absolutely necessary for the message of the gospel and salvation? Is the gospel necessarily predicated on the practice of slavery?

When skeptics point to the practice of slavery in the Bible as an example of something disagreeable, I will most certainly agree with the author that these skeptics are often doing so based on a very modern sense of morality which finds the very notion of slavery to be completely abhorrent, and which has been fostered by the still-fairly-recent history of horrific slavery in modern nations like the United States of America.

So what?

If one’s goal is to argue that these detractors are mistaken to condemn the Bible for its sanctioning of the practice of slavery, then there are only two real ways in which this can be done. Either one must argue that there are situations in which slavery is a perfectly ethical practice or else one must argue that the Bible does not establish laws and processes by which slavery could be instituted.

Despite Todd Clay’s assertions, God’s existence is still not obvious to me. In fact, the arguments which he presents are quite bad. Indeed, it seems to me that he is claiming to have found Jesus in his philosophical toast.

The Fine Tuning Argument:Scientists have determined that life in the universe would be impossible if more than about two dozen properties of the universe were even slightly different from what they are. In short, the universe is “fine-tuned” for life.

The universe has properties which account for why the universe is the way it is. This does not imply, in any way, that these properties could have been any different than they are. Even if they could have been different, that in turn does not imply that those properties were– or even could have been– deliberately designed by an intelligent being.

2.

Earth’s Perfect Size:Earth is the only known planet with an atmosphere of the right mixture of gases to sustain plant, animal and human life.

This is not a different argument, but rather a specific example of one of the properties mentioned in #1. Nothing in this datum implies anything at all about the existence of deity.

3.

Irreducible Biological Complexity:There are a number of biochemical systems that are irreducibly complex. Like the functions of a mousetrap, a cilium cannot perform its function unless its components are arranged and structured in precisely the manner in which they are.

The concept of “irreducible complexity” is immensely popular among Christian apologists. It is, however, not a concept which holds any sort of weight in biology scholarship. Regarding the specific case mentioned, there have been numerous discussions from biologists regarding the manner in which cilia could have evolved. While it is true that definitive and conclusive answers to the issue are elusive, the fact that it remains an open question does not imply that the answer is that they therefore cannot have evolved. To assert otherwise is a blatant *argumentum ad ignorantiam* fallacy.

4.

DNA’s Function as Life’s Blueprint:DNA instructs the cell like your smartphone is programmed. DNA is a three-billion-lettered program telling the cell to act in a certain way. It is a full instruction manual for life.

DNA is a molecule. Just like every other molecule, DNA is a composition of atoms which interact with one another according to well-understood processes of physical chemistry. It is simply untrue to claim that “DNA instructs the cell like your smartphone is programmed.” Despite talking points from apologists, the processes are highly dissimilar. They bear a superficial resemblance which allows software programming to stand as an interesting metaphor for DNA’s biological function. However, it’d be a mistake to think that the metaphor therefore implies that DNA must share *all* of its properties with software programming.

5.

The Human Brain’s Capacity:The human brain processes more than one million messages a second. Your brain weighs the importance of the data and allows you to focus and operate effectively in your world.6.

The Eye’s Complexity:The human eye has automatic focusing and handles an astounding 1.5 million messages simultaneously. In addition, it can distinguish between seven million colors making it a highly complex organ.

I fail to see how either of these has anything to do with whether or not deity exists, at all.

7.

Chaos Never Creates Order:Since everything in the universe tends toward disorder, it is a violation of the Second Law of Thermodynamics if things become more orderly and complex by themselves. Hence, something must have put everything together originally.

Contrary to what Christian apologists seem believe, the Second Law of Thermodynamics does not say, “everything in the universe tends toward disorder.” Such a statement is rather obviously untrue, and as such would make for a fairly ridiculous law of physics. The actual Second Law of Thermodynamics states that the entropy of a closed system is unlikely to decrease and rather tends to increase or remain steady. “Entropy” is not a measure of disorderliness. Entropy can decrease over time in an isolated system; it is simply unlikely that it will do so. Furthermore, in an open system, entropy can quite easily decrease over time. Local regions of the universe, such as our little planet, are most certainly open systems.

8.

Quantum Mechanics Analogy to God:John Polkinghorne suggests the nearest analogy to the existence of God in physics is the ideas of quantum mechanics. This field makes sense of a great deal of disparate data.

This statement is, at best, curious. At worst, it is utter nonsense. In what way is quantum mechanics analogous to God? Simply because it makes sense of a great deal of disparate data? So does physical chemistry, and stellar mechanics, and Relativity, and literally every other field of physics. Presumably, Mr. Clay has omitted every part of John Polkinghorne’s claim which is actually relevant and meaningful.

9.

Human Consciousness:The workings of human consciousness are similarly miraculous. Consciousness has no physical presence in the world; the images and thoughts in our consciousness have no measurable dimensions which begs the question where it actually resides.

This claim would appear quite silly to a neuroscientist. Human consciousness can be measured in numerous different ways– for example, through electroencephalography or functional magnetic resonance imaging. Memories, which Mr. Clay mentions specifically, are quite well known to be intrinsically linked to the hippocampus. Damaging the hippocampus can lead to a loss of memory, or even to the loss of the ability to form new memories. Furthermore, physical changes to other parts of the brain– including simply altering the amounts of certain chemicals in the system as well as actual damage or structural alteration– is well known to have significant effects on the way a person thinks. Quite the contrary to Mr. Clay’s assertion, everything we’ve learned in neuroscience seems to assert that consciousness most certainly is a physical process.

10.

The Transcendental Argument:

– Logical absolutes (Laws of Logic) exist.

– Logical absolutes are conceptual by nature–are not dependent on space, time, physical properties, or human nature.

– They are not the product of the physical universe (space, time, matter) because if the physical universe were to disappear, logical absolutes would still be true. Logical Absolutes are not the product of human minds because human minds are not absolute.

– Since logical absolutes are always true everywhere and not dependent upon human minds, it must be an absolute transcendent mind that is authoring them.

– This mind is called God.

There’s quite a bit wrong with this argument. Unless one subscribes to a Platonist philosophy, it is not the case that logical absolutes “exist” in the same way that, for example, the sun “exists.” In reference to concepts (such as “logical absolutes”), the phrase “exists” means that the concept is descriptive of something true, in reality. It does not mean that the referent has a distinct instantiation in reality. It is not the case that such concepts exist apart from the universe, as they are– in fact– descriptive of the universe.

If, on the other hand, one does subscribe to a Platonist philosophy, then concepts such as logical absolutes do have distinct instantiation in reality, and that instantiation is not dependent upon anything else. There is no need for a mind to author them in order that they exist.

11.

Argument From Truth:Our limited minds can discover eternal truths about being. Truth properly resides in a mind. But the human mind is not eternal. Therefore there must exist an eternal mind in which these truths reside.

“Truths” are concepts, just like “logical absolutes.” As such, the same discussion applies here. There is simply no way to logically imply the existence of an “eternal mind” from the fact that we can describe things– even “eternal” things– about the world.

12.

The Laws of Math:It takes the existence of some kind of a God to make the mathematical underpinnings of the universe comprehensible.

Preposterous. There is nothing about mathematics which necessitate “the existence of some kind of a God.” The comprehensibility of mathematics is expressly due to the fact that human beings created mathematics to be comprehensible. It would not make much sense to create a system to describe the world which was incomprehensible.

13.

Ontological Argument:Anselm argued that God was a being who possesses all conceivable perfection. But if this being “existed” merely as an idea in our minds, then it would be less perfect than if it actually existed. So it wouldn’t be as great as a being who actually existed, something that would thus contradict our definition of God — a being who’s supposed to be all-perfect. Thus, God must exist.

This argument wasn’t even convincing to other Christians when Anselm first offered it. One cannot simply define a thing into existence. For example, if I were to say, “Dragons are defined as existent magical creatures,” no one would therefore conclude that dragons must exist. Similarly, one cannot simply assert that a definition for God which includes existence therefore implies that God exists.

14.

Cosmological Argument (First-Cause Version):– Things exist.

– It is possible for those things to not exist.

– Whatever has the possibility of non-existence, yet exists, has been caused to exist. Something cannot bring itself into existence since it must exist to bring itself into existence, which is illogical.

– There cannot be an infinite number of causes to bring something into existence. An infinite regression of causes ultimately has no initial cause, which means there is no cause of existence. Since the universe exists, it must have a cause.

– Therefore, there must be an uncaused cause of all things.

– The uncaused cause must be God.

I think we can all agree that things exist. We can also agree that it is possible for at least some of those things to not exist. However, it is far from clear that it is possible for *all* things to not exist. Nor is it clear that something which has the possibility of non-existence has been caused to exist. Nor is there any good reason to think that there cannot be an infinite regress of causation. Faulty premises piled upon one another do not make for a very convincing argument.

15.

Contingency Argument (Something Rather Than Nothing):– Everything that exists has an explanation of its existence.

– If the universe has an explanation of its existence, that explanation is God.

– The universe exists.

– Therefore, the explanation of the universe’s existence is God.

This is quite obviously a fallacious attempt to simply presume one’s conclusion. No justification is offered for the claim that the explanation of the universe must be God. A person could just as easily and just as accurately replace the word “God” in the above argument with the word “cheese.” I doubt anyone would agree that this logically implies that cheese must therefore be the explanation of the universe’s existence.

16.

Teleological Argument (Design):Every complex human creation (like a painting or a jumbo jet) demands we explain it’s existence through an intelligent, skillful creator – not random processes. The universe is billions of times more complex than any human creation. Therefore a creator made it: God.

Yes, every complex human creation is a thing created by a human. That’s a fairly uninformative tautology. It is the fact that it is a human creation which causes us to draw the conclusion that it was created by a human, not the fact that it is complex. Complexity does not imply the necessity of design.

17.

Argument from Change:If there is nothing outside the material universe, then there is nothing that can cause the universe to change. But it does change. Therefore there must be something in addition to the material universe.

This does not follow logically, at all, nor is it even coherent. The word “change” describes different spatial states over successive moments of time. Both the spatial states and the moments of time, in question, are subsets of the universe. The concept of change doesn’t even make sense when considering the universe as a whole.

18.

Argument from The Mind:Philosopher J. S. Mill summed up where we have now come to: “It is self-evident that only Mind can create mind.” The only reasonable conclusion is that an eternal Creator is the one who is responsible for reality as we know it.

There is nothing, at all, self-evident about the claim that “only Mind can create mind.” It’s not even clear what it is supposed to mean. I see no reason at all to think that biological chemistry cannot produce a mind. Quite the contrary, it seems that every mind with which I am familiar has been produced by biological chemistry, and not with any apparent direct guidance from a mind.

19.

Moral Argument:All people recognize some moral code. Right and wrong imply a higher standard or law, and law requires a lawgiver. Because the Moral Law transcends humanity, this universal law requires a universal lawgiver. This lawgiver is God.

This argument tends to skip over the fairly obvious: if the disparate moral codes followed by human beings have been created, what reason is there to think that they have not been created by human beings? There’s no reason to invoke deity.

20.

Existence of Evil:A corollary to the Moral Argument: the problem of evil presents a question not only for the person trying to give an answer to the problem, but also for the questioner. The problem is that it presupposes a standard of goodness, and that presupposes a design plan. If true, these objections then both point to God whose character is the very standard of goodness.

This is a complete misunderstanding of the Problem of Evil. Yes, the Problem of Evil presupposes a standard of goodness. It does so in order to show that a position under discussion is self-contradictory, inconsistent, and irrational. Christians claim that there exists a God who is omnipotent and omnibenevolent, with his benevolence being according to some universal and transcendental standard. The Problem of Evil presupposes those premises in order to attempt to demonstrate that these things are incompatible. In one of his famous dialogues, Plato had the philosopher Euthyphro voice this incompatibility with a widely known dilemma. If God wants to stop evil but cannot do so, then God is not omnipotent. If God can stop evil but does not want to do so, then God is not omnibenevolent.

21.

Argument from Miracles:A miracle is an event whose only adequate explanation is the extraordinary and direct intervention of God. There are numerous well-attested miracles. Therefore, there are numerous events whose only adequate explanation is the extraordinary and direct intervention of God. Therefore God exists.

This, like the Ontological Argument from #13, is yet another attempt to simply define God into existence. Yes, there are numerous events which have occurred which people claim to be miracles. That does not therefore imply that the only adequate explanation for any of these events is the extraordinary and direct intervention of God.

22.

Argument from Beauty:Peter Kreeft explains Von Balthasar argument: “Beauty reveals God. There is Mozart, therefore there must be God.”

This argument is entirely nonsensical. Beauty is a completely subjective quality. One might as well claim, “Putridness reveals God. There is vomit, therefore there must be God,” or even, “Beauty reveals four-sided triangles. There is Mozart, therefore there must be four-sided triangles.”

23.

Argument from Degrees of Perfection: If degrees of perfection pertain to being and being is caused in finite creatures, then there must exist a “best,” a source and real standard of all the perfections that we recognize belong to us as beings. This absolutely perfect being—the “Being of all beings,” “the Perfection of all perfections”—is God.

The fact that we have standards which define the perfect example of a particular quality does not imply that there must therefore exist something which instantiates that perfection. We do not need an actual example of a perfect circle which exists in reality in order to coherently compare something to a perfect circle.

24.

Argument from Religious Experience: Many people of different eras and of widely different cultures claim to have had a “divine” experience. It is inconceivable that so many people could have been so utterly wrong about the nature and content of their own experience. Therefore, there exists a “divine” reality.

It is entirely conceivable that so many people could have been so utterly wrong about the nature and content of their own experience. Many people of different areas and of widely different cultures claim to have experienced the sun circling around an unmoving Earth. Surely we can agree that it is conceivable that so many people could have been so utterly wrong about the nature and content of their own experience. Why should one think that claims about the divine are any different?

25.

Other Minds Analogy:Technically not a theistic proof, this analogy illustrates how difficult it is for a determined atheist to apostatize. Alvin Plantinga compares the question of the existence of God to the question of the existence of other minds, claiming both are notoriously impossible to “prove” against a determined skeptic.

These two positions are not analogous. In fact, they are complete opposites. The solipsist thinks that all which is observable is not actually real. On the other hand, the atheist rejects the claim that a particular thing which is completely unobservable– that is, God– is real. I daresay Alvin Plantinga rejects claims that land wights exist. Would he consider his own rejection of that claim to be analogous to solipsism?

26.

The Historicity of the Old Testament:“The historicity of the OT should be taken seriously. As for the OT text itself, the Dead Sea Scrolls (ca 150 b.c.- a.d. 70) provide good evidence of a carefully transmitted core-text tradition through almost a thousand years down to the Masoretic scribes (ca eighth-tenth centuries a.d.) Thus, the basic text of OT Scripture can be established as essentially soundly transmitted, and the evidence shows that the form and content of the OT fit with known literary and cultural realities of the Ancient Near East.”

I agree that the historicity of the OT should be taken seriously. I also agree that the OT has been carefully– though certainly not perfectly– transmitted through millennia. However, it does not follow from the fact that the text has been reliably transmitted that the text therefore recounts historical facts.

27.

The Historicity of the New Testament: “Historiography is a branch of study which focuses on the logical, conceptual, and epistemological aspects of what historians do. Critical historiography studies, among other things, the different tests which should be applied to a document to determine whether or not it is historically reliable. When many of these tests are applied to the New Testament documents, they show themselves to be as reliable as, or superior to, most other ancient documents.”

Well, considering “most other ancient documents” includes quite a bit of legendry, mythology, and fiction, this isn’t really saying much. However, even if I were to assume the author meant to specifically compare the NT to other ancient histories, this still isn’t really that bold a claim. There are quite a number of things in other ancient histories which modern historians strongly doubt have any basis in actual history. Epimenides, Eudoxus, and Xenocrates all report that a woman was impregnated by the god, Apollo, in order to give birth to Pythagoras. From other historians, we have reports that Pythagoras had a thigh made of solid gold, that he flew great distances by means of a magic arrow, that he appeared in two different cities at the same time, and that a river greeted Pythagoras verbally, by name.

Pythagoras is just one of thousands of examples. Ancient histories are replete with legendary accounts which no modern historian would accept as being actually historical.

28.

Survival of the People of Israel:French philosopher Blaise Pascal wrote at length about the marvel of Jewish survival. Powerful kings have tried to destroy them, yet the Jews survived whereas the nations of Greece, Italy, Athens and Rome have long perished. It is said that when King Louis XIV of France asked Pascal to give him proof of God’s existence, he replied, “Why, the Jews, your Majesty ― the Jews.”

The fact that a particular culture was able to survive huge amounts of adversity does not, in any way, imply that the religious beliefs of that culture accurately represent reality. In fact, if this was the case, it would pose some rather serious problems for Christianity.

29.

Biblical Miracles: “We have now studied more than 35 miracles described in the Bible, and many more could be discussed. These were done by God at the hands of at least ten different prophets. They were recorded by at least ten Bible writers in at least 15 books of the Bible. In addition, the accounts cite more than 15 individuals or groups who were opponents of God’s people but who directly or indirectly acknowledged that Bible miracles really did occur. In addition another 15 individuals or groups (who were not necessarily opposed to the message) are also cited as having acknowledged Bible miracles. That makes a total of more than forty individuals or groups who admitted Jesus or Bible prophets really did miracles.”

The fact that some people accept– or, at the very least, do not outright reject– claims regarding certain events does not imply that those events are historically accurate. Once again, Pythagoras was claimed by numerous people to have performed many miracles, and those claims were believed by countless more people. Does that imply that Pythagoras actually performed those acts?

30.

Fulfilled Messianic Prophecies:Jesus was the promised Messiah from the Old Testament (or Tanach). There were hundreds of prophecies he fulfilled including that he would be the seed of a woman (Genesis 3:15), a descendant of David (2 Samuel 7:12-16), called God’s son (Psalm 2:1-12), and many more.

It’s quite easy to try to retroactively shoehorn a leader into vague prophetic literature. Look! Genesis 3:15 says that a descendant of Eve will oppose the evil serpent! Well, Jesus is a human being and Jesus opposes evil! No one else could POSSIBLY fit that description. Oh, the Messiah is supposed to be born of the line of David? Well, here’s a genealogy showing that David was one of the ancestors of Joseph, father of Jesus. Not convincing enough? What if I gave you a *second* genealogy of Joseph, one which is entirely different from the first except that it also says David was one of his ancestors? The Messiah will be called God’s son! It’s absolutely impossible that anyone other than the Messiah could be called God’s son– except for all those hundreds of other people from numerous different cultures who have been called God’s son, of course.

Oh, but all those prophecies about how the Messiah is supposed to topple the political oppressors over the Jewish people so that they might re-establish an earthly kingdom? It doesn’t matter that Jesus of Nazareth didn’t fulfill any of those.

31.

The Resurrection of Jesus:The Apostle Paul once implied that Christianity was worthless if Jesus didn’t rise from the dead (1 Corinthians 15:14). Brian Chilton outlined 10 reasons why Jesus actually rose from the dead including how the first eyewitnesses were women (a woman’s testimony was not admissible in court in the first century), embarrassing details about the resurrection, and the disciples willingness to die believing in the. (sic)

The claim that a woman’s testimony was not admissible in court in the first century is a very common lie, but a lie nonetheless. It is true that it was considered rude or unseemly to ask a woman to appear before a court unless absolutely necessary; however, in those times that it was necessary for a woman to testify, her testimony was not trusted any less than that of a man. Cicero famously shamed Gaius Verres after the famous Roman orator had to ask several women to testify against Verres in court. The testimony of those women was not doubted on account of their gender. Outside of the courtroom, Josephus relied on the testimony of women when compiling his historical accounts of Gamala and Masada. Even in the New Testament, itself, the testimony of women is not only trusted, but quite often lauded. John 4:39 recounts that a great many Samaritans came to believe in Jesus on the basis of a woman’s testimony. Paul’s letters name quite a number of women who were well respected and whose testimony was presumably trusted, particularly in Romans 16 where he mentions a number of woman who were teachers and leaders in the Church– Phoebe, Prisca, Mary, Junia, Julia, and the sister of Nereus. In particular, he explicitly states that Phoebe is a deacon of the church at Cenchreae, and that Junia was a prominent apostle (Gk., ἐπίσημοι ἐν τοῖς ἀποστόλοις).

As for the disciples’ willingness to die for their belief in (presumably) the Resurrection, even if we accept that this was the case this does not imply that their belief must therefore reflect a true fact about reality. Many people are willing to die for sincerely held beliefs which are, nonetheless, wrong. With all that said, I see very little historical evidence to believe that the disciples were willing to die for this belief, in the first place. Even in the New Testament, the only person explicitly threatened with death for his belief in the Resurrection was Stephen, a deacon in Jerusalem and not a disciple who had witnessed Jesus’ death and Resurrection. James, son of Zebedee, was executed by Herod, and Peter was arrested around the same time, but the specifics of their offense are left to the reader’s imagination. Paul was threatened with arrest for inspiring Jews to abandon ritualistic Law, not for preaching that Christ was risen. The purported martyrdom of each of the Twelve (besides James bar Zebedee and ignoring Judas Iscariot) are all later traditions, formed well after the New Testament had been written, and each is historically dubious. There is no good reason to believe that all of the men who purportedly witnessed the risen Jesus died or were even threatened with death for that belief.

Christians and atheists, alike, quite often laugh at those people who claim that the image of Jesus appeared on their breakfast or in a random stain or in some other odd pattern. The arguments which Todd Clay highlights in his blog article are the philosophical equivalent of this. These arguments jump from vagaries or illusory patterns to the claim that God must therefore exist.

There are better arguments for the existence of God (better in that they are more well-formed, though not necessarily any more convincing than the ones Mr. Clay presents). Some of those arguments have been the subject of discussion on Boxing Pythagoras in the past. I’ve chosen to discuss Todd Clay’s article, today, because it is a very good example of the manner in which talking points often get passed from one apologist to the next like a theological game of telephone, to the point where the person spouting them off has not likely spent much time actually thinking them through, let alone conversing with those who might oppose such positions. They are presented as if they are cold, hard truths and brute facts of reality when in fact they are highly contentious and extremely debatable.

Even if you are the type of Christian that believes the Bible to be the infallible Word of God, you most certainly should not think that Christian apologists are similarly infallible. Question their arguments. Think them through. Are they truly even valid arguments? Do they use logical rules of inference to proceed from assumptions to valid conclusions? If they do, are those assumptions questionable? What counter-arguments might be brought against them? Do not simply regurgitate the half-understood positions which have been presented to you by other people who happen to have similar beliefs.

Stop trying to tell me that your philosophical toast looks like Jesus.

]]>Mathematicians are not most people.

For quite a long time, now, mathematicians have recognized that there are at least two very distinct ways in which we use numbers to describe things. Being the scholarly, academic types that they are, mathematicians have assigned names to these two different types of numbers which sound heady and difficult to the average person: *ordinal* numbers and *cardinal* numbers. Indeed, even mathematics students sometimes need quite a bit of work and explanation in order to really grasp the difference between these two types of number; but I’m going to do my best to explain these things in a very simple way for a casual audience.

Ordinal and cardinal numbers roughly correspond to the ideas of *value* and *size*, respectively.

Everyone can fairly intuitively recognize that value and size are entirely distinct concepts. If I have a bar of lead and a bar of gold which are both 30 cubic centimeters, it doesn’t take much to realize that both are the same size. However, if I were to allow you to choose one to take, who would hesitate to grab the gold bar? Despite being equal in size, the two bars are certainly not equal in value. The gold has far, far greater value than does the lead. What most people do not realize is that there is a similar concept to be found among numbers.

In order to explain, I’m going to ask you to take a look at the picture below.

In this picture, we have a set of ravens and a set of owls. Without counting or using numbers to describe either set, how can we know that both sets are the same size? Most people will immediately intuit that this is the case, even before recognizing that there are 5 of each, ravens and owls. However, those same people usually don’t understand why. You see, our brains tend to naturally make a connection that we might not consciously recognize at first. We notice, without even realizing it, that for each raven there is also an owl. We can match them up in what mathematicians refer to as a one-to-one relationship, as in the image below.

When the elements of two sets can be placed into a one-to-one relationship, like this, we say that they have the same *cardinality*. If they cannot, then the cardinality of the two sets is different. As such, cardinality is a means of discussing the size of a set.

However, as mentioned, there is another way in which we think about numbers, too. Have a look at the following image. In it, you’ll see a series of symbols being compared to one another using less-than symbols (<). Again, without assigning any numbers to the symbols, can you place them in order from least to greatest?

It’s not a very difficult puzzle. We see that the Mirror is less than the Mask, but the Wolf is less than the Mirror. Similarly, the Rose is less than the Scepter, but the Mask is less than the Rose. All it takes is a little rearrangement of the information which we have been given to see the following:

Without using any numbers, at all, we’ve placed these unfamiliar symbols into order from least to greatest. We don’t need any numbers to know, for example, that Wolf is less than Mask or that Mirror is less than Scepter. In short, we’ve established an order of value for these symbols, without needing to know anything else about them, at all. This sense of ordering things by their value is what we mean when we discuss *ordinal* numbers.

Thus far, everything we’ve discussed should seem fairly simple and straightforward. You might, in fact, wonder why I’ve even chosen to discuss it, at all. Sure, ordinality and cardinality might be two different concepts, but what difference does it really make? One does not need to consciously understand these differences in order to talk about numbers in a meaningful way.

Indeed, when we are discussing the numbers with which most people are quite familiar, saying that ordinal numbers are different than cardinal numbers seems like a distinction without a real difference. Who cares if we are talking about the cardinal number 5 or the ordinal number 5, for instance? Isn’t the number 5 just the number 5 regardless of whether we’re talking about ordinality or cardinality?

In the case of finite sets, this holds quite well. The cardinal 5 and the ordinal 5 aren’t really very different, other than in some minor quality about how we use them. The whole notion seems like quibbling pedantry, to the layperson. The problem arises, however, when we begin talking about infinite sets. It does not take very long to realize that with infinite sets, ordinals and cardinals are very different things.

Let me explain by discussing one of the ways in which we can construct the Natural numbers. We begin with an empty set– that is, a set which contains no elements. Now, we can play a little game with one very simple rule. The rule is that we are allowed to create a new set called a Successor which contains as its elements every other set which we already have. Since we currently only have the empty set, this means that we can create a new Successor set which contains the empty set as its only element. Once we have done that, we can apply the rule again, this time creating a Successor set which contains the empty set and the set containing the empty set. We can then apply the rule again to find another Successor, and we can continue this process ad infinitum.

If this seems confusing, take a look at the image below. The boxes represent our sets, while commas separate the elements within a set.

Now, all this “Successor to the Successor to the…” language is the confusing part of all this, so let’s make things easier by assigning symbols to represent each of the sets which we have created. We can assign the symbol “0” to the Empty Set, the symbol “1” to the Successor to 0, the symbol “2” to the Successor to 1, and the symbol “3” for the Successor to 2. I’m sure you now see where I am going with this.

If we continue playing our Successor game, we can probably already see that the set “4” will contain 0, 1, 2, and 3. The set “5” will have sets 0, 1, 2, 3, and 4 as its elements. The set “6” will be constructed of elements 0, 1, 2, 3, 4, and 5. Et cetera, et cetera. This is all very straightforward. All of the numbers which are created in this way are termed the Natural numbers by mathematicians.

This setup gives us a clear method for discussing the *ordinality*, or value (as I have been describing it), of a set. A set is said to be “less than” another set if the former is an element of the latter. So, 1 is less than 3 because 3 contains 1 as an element, in this construction. Similarly, 0 is less than all of the other sets because 0 is an element of all of the other sets. Conversely, 2 is not less than 1 because 2 is not an element of 1; and no Natural number is less than 0 because no Natural number is an element of 0.

We can also use this method to define what it means for sets to be *ordinally equal*. If we have two sets, A and B, we can say that A and B are *ordinally equal *if A is not less than B and B is not less than A. So, for example, if we were to compare our set “3” to a copy of itself, we would see that original-3 is not less than copy-3 (because the set “3” does not contain 3 as an element), and copy-3 is not less than original-3 for the same reason. Therefore, we would conclude that 3 is ordinally equal to 3.

We can also use this setup as a very easy way of discussing the *cardinality*, or size, of a set. For an illustration of this, take a look at the next image.

Let’s say that we want to compare the set which we called “3” with the set of owls, pictured here. If you will recall from our discussion of ravens and owls earlier, if the elements of two sets can be placed in a one-to-one relationship with one another, then those two sets have the same cardinality. So our set of owls, in this picture, has the same cardinality as our set “3” has because we can match each of the owls to each of the elements in 3. When two sets have the same cardinality we can say that they are *cardinally equal*.

With the Natural numbers, it is plain to see that two sets which are *ordinally equal *are always going to be *cardinally equal,* as well. The set 3, for example, is not less than itself and it can have its elements matched one-to-one with itself. Similarly, if two numbers are not ordinally equal they will not be cardinally equal either. So, 3 is less than 4, and there is no way to match up all of the elements of 3 and 4 so that they will be in a one-to-one relationship.

This all may seem rather obvious to some. Of course two sets which are ordinally equal are going to be cardinally equal! How could things be any other way? One might wonder anyone would bother to worry about the distinction between ordinality and cardinality, at all. The truth is, however, that numbers are far stranger than most people will ever realize.

Let’s say that I create a new set which contains as its elements all of the Natural numbers. We will call this set “ω” (the Greek letter *omega*). By the definition of “less than” which we used earlier, it is clear that every Natural number is less than ω, since every Natural number is an element of ω. Similarly, we see that ω is not less than any Natural number, since no Natural number would have ω as an element. We have created a completely new ordinal number.

Now, we can apply our Successor rule again to create a set which contains all of the Natural numbers and ω as its elements. We’ll call this Successor set “ω+1.” Clearly, since ω is an element of ω+1, we can see that ω is less than ω+1. Again, we can continue to apply the Successor rule to create new sets which we might call “ω+2,” “ω+3,” “ω+4,” and so on, and we can continue to see that our ordinal “less than” rule still holds for each of these. We call ω and the Successors which can be generated from it *transfinite* ordinals.

One might easily think that, just as was the case with the Natural numbers, if two transfinite sets are not ordinally equal, then they can’t be cardinally equal, either. But, as we shall see, the world of the infinite is a very, very different place.

Now, it’s clear that ω does not have the same cardinality as any of the Natural numbers, because no matter which Natural number one chooses, attempting to set up a one-to-one relationship between its elements and those of ω will result in unmatched elements remaining in ω; and given all the precedent which we saw in the Natural numbers, we might expect that ω+1 has a different cardinality than does ω. Indeed, at first glance this might seem to be the case. When we try matching the elements of ω to those of ω+1, the obvious thing would be to match 0 to 0, 1 to 1, 2 to 2, and so on. If we were to go about it in this way, we would match up every Natural number in the set ω to its counterpart in ω+1 and there would still be one unmatched element in ω+1.

However, what if we were to be a little bit clever? Instead of matching each Natural number to its counterpart in the other set, let’s instead match the ω-element of ω+1 with the 0-element of ω. We can then match each Natural number element in ω+1 with its Successor element in ω, so that 0 matches with 1, 1 matches with 2, et cetera. Now, we find every element in ω+1 *does* pair up with an element in ω in a one-to-one relationship.

Despite the fact that ω is ordinally less than ω+1, we can see that these two sets are cardinally equal! In fact, ω has the same cardinality as *any* of the Successor sets which can be generated from it. So, ω has the same cardinality as ω+1 or ω+15 or ω+3672. Mathematicians refer to the cardinality of ω with another symbol, (pronounced “Aleph null” or “Aleph zero,” from the Hebrew letter). So, despite the fact that the ordinal number 3 and the cardinal number 3 don’t seem to be very different, it becomes exceedingly clear that the ordinal number ω and the cardinal number are not the same thing, at all.

A bar of lead and a bar of gold might be the same size, but the bar of lead has far less value than the bar of gold. Similarly, we now see that it is entirely possible for two sets to be the same size, even if one has far less value than the other. Though we do not usually see it in our everyday experience, there is a very distinct and meaningful difference between numbers which are used ordinally and numbers which are used cardinally. Despite the fact that the concept is so thoroughly ingrained in us, it is not always obvious what it is that we mean when we use numbers.

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The First Way is Aquinas’ Argument from Motion. The philosopher, being heavily entrenched in Aristotelian philosophy, means something different by “motion” than is commonly understood, today. When Aquinas says that a thing is “in motion,” he means that some change which has yet to occur is being made in that thing. In addition to spatial mobilization, this can refer to any sort of change—for example, a chameleon which is perfectly still, but changing colors, would be “in motion,” on Aquinas’ view. We can summarize this argument as follows:

- There are some things which are in motion.
- If a thing is in motion, that motion must originate in the action of some other thing.
- This chain cannot go to infinity, because there must be a First Mover.
- Therefore, there is a First Mover whose motion does not originate in the action of some other thing. (3)

Laid out in this fashion, it seems perfectly clear that Aquinas’ First Way is not even a valid argument, let alone a sound one. His third premise is a blatant instance of question begging. Aquinas asserts that there must be a First Mover, therefore there is a First Mover. Furthermore, the conclusion which he draws violates his second premise, making it a fairly clear example of special pleading. If Aquinas’ conclusion is true, then his second premise is false, invalidating the argument.

Aquinas’ dedication to Aristotle is again evident in his Second Way, the Argument from the Formality of Efficient Causation. Aristotle’s view of the notion of causation divided the concept into four categories: material, efficient, formal, and final. The “material” cause is that which is being changed, the “efficient” cause is the agent effecting the change, the “formal” cause is the manner in which the change occurs, and the “final” cause is the end purpose which drove the change (Aristotle’s *Physics* 2.3). Using an argument which is similar to his First Way, Aquinas argues:

- Changes have Efficient Causes.
- No change can be its own Efficient Cause.
- There cannot be an infinite chain of Efficient Causes.
- Therefore, there must be a First Efficient Cause. (1, 2, 3)

This argument is slightly better than Aquinas’ First Way, as this one is at least valid and free of overt logical fallacies. However, it is not necessarily sound. The third premise is fairly suspect, and Aquinas does not give very good support for it. The priest notes that in any series of efficient causes, a subsequent efficient cause is always the result of a prior. He then baldly asserts that there must be a first Efficient Cause. This, again, shows Aristotle’s influence on Aquinas, as the Greek philosopher was vehemently opposed to the notion of actually infinite sets of entities (*Physics* 6.1). Unfortunately for Aquinas, Aristotle’s understanding of infinity is nearly as antiquated as was his belief that the Sun, planets, and stars revolve around the Earth. Mathematicians and physicists have been regularly utilizing completed infinite sets in real-world applications for more than 400 years, now, and it has been nearly 100 years since the concept has even been a source of much controversy in the philosophy of mathematics. If Aquinas wants to demonstrate that there cannot be an infinitely receding chain of Efficient Causes, he must do a better job than to simply note the fact that this would imply that the chain lacks a first cause.

The Third Way is an Argument from Contingency. Aquinas points out that there are things which exist that exhibit the potential to not exist. Four centuries later, in a similar argument, Leibniz would say that such things are “contingent,” as opposed to “necessary” things—that is, things which cannot logically be non-existent. Aquinas’ Third Way is summed up as follows:

- Some things exist which have the potential to cease existing.
- If a thing can cease to exist, then there must have been a prior time in which it did not exist.
- If everything has the potential to cease existing, then there must have been a time when nothing existed. (1, 2)
- If nothing existed, nothing can have been brought into existence.
- Therefore, it cannot be that everything has the potential to cease existing, since (4) contradicts with (1).

There are two problems with this Third Way which are of particular concern. Firstly, Aquinas offers absolutely no justification for his second premise. It does not follow that a thing potentially ceasing to exist implies that it must have not existed at some time in the past. However, even if that premise is true, the most which Aquinas might draw from this argument is that there must exist *at least* one thing which is necessary. He has not demonstrated that there can only be a single necessary thing. Aquinas gives no reason to think that there might not be a multitude of necessary things which do not have their necessity caused, but rather exist necessarily in and of themselves. Certainly, if numerous such things were in existence, Aquinas would not say that the whole set of them is what is meant by the name, “God.”

Aquinas proposes an Argument from Gradation as his Fourth Way. He discusses the manner in which things are described, noting that some things exhibit a certain property to a greater or lesser scale than do others. He claims that this gradation is judged in comparison to some singular exhibitor which most exemplifies that property.

- There are some properties of things which can be accorded as greater or lesser.
- A property can only be accorded as greater or lesser as it resembles a thing which is the greatest exhibitor of that property.
- “Being,” “goodness,” and every other perfection can be accorded as greater or lesser in a thing.
- Therefore, there must exist a greatest exhibitor of “being,” “goodness,” and every other perfection. (1, 2, 3)

The second premise of this argument is worse than just dubious. Indeed, it would seem to be glaringly false. In his own example supporting the premise, Aquinas states that “hotter” is judged in comparison to fire, which must be considered “hottest,” since it is the source of all heat– yet another antiquated and abandoned position which Aquinas acquired from Aristotle. However, it is now known that fire is *not* the source of all heat, and that there exist things which are far, far hotter than fire. Even more damning to this perspective, however, is that a person need not ever know anything of fire to know that a stone which had been left out in the sun is hotter than one left in the shade. Nor is it even clear that “hottest” is a notion which can be applied to any single, universal thing. The gradation of a property is not judged by comparison to the maximal extremes of that property, but rather by comparison to other things which exhibit that property. An oven is hotter than a refrigerator and a blast furnace is hotter, still, but these are all still exceptionally cold in comparison to the surface temperature of a star.

Finally, the Fifth Way is an Argument from Teleology. Aquinas claims that there exist things which lack intelligence, and which yet act to a purposeful end. This purpose, he claims, must be instilled into those things by an agent with intelligence.

- There exist things which lack intelligence and act for some purpose.
- A thing which lacks intelligence cannot act for a purpose unless it is directed to do so by a being with intelligence.
- Therefore, some intelligent being exists by whom all things which lack intelligence are directed to their purpose. (1, 2)

Once again, Aquinas’ formulation seems problematic. Firstly, it is not clear that things which lack intelligence are acting in accordance with some distinct purpose. His attempted justification for this point does not seem to really lend it much support. Aquinas claims that these things act “always, or nearly always, in the same way, to obtain the best result.” It is far from demonstrated that this is the case; however, even if it is, it does not follow that that their actions are therefore directed by an intelligent being. Aquinas simply asserts that this is so. Again, this would seem to be a function of the priest’s reliance upon Aristotle, and his ignorance of the physics which would be discovered centuries after his *Summa*. Furthermore, as with the Third Way, Aquinas gives no good reason to think that all of these natural, intelligence-lacking things must therefore be directed by a *single* entity, other than the earlier argument regarding Efficient Causation which has already been seen to be problematic. Even if his premises were true, Aquinas is not justified in this leap of logic. The planets, for example, might each be driven in their orbits by an intelligent being distinct from that driving any of the others, and neither of the first two premises would be violated. As before, it is doubtful Aquinas would agree that all intelligent beings are what is meant by the name, “God.”

The Five Ways of Thomas Aquinas’ *Summa Theologica* are, to be certain, an important part of the history of philosophy. However, their brilliance and validity are very often overstated by proponents. Aquinas’ views are mired in a foundation of the positions of Aristotelian physics which, though extremely influential in 13^{th} and 14^{th} Century Christendom, are now extremely antiquated and which have been abandoned by all serious scholarship. Even within that framework, however, his arguments are rather loosely proffered and lacking in rigor. Aquinas’ Five Ways do not exhibit the strength of argument with which they are commonly characterized. As such, the question of God’s existence does not seem to be quite so well founded as the Italian priest claimed.

Joseph Nebus has recently written a couple of posts (here and here) in which he discusses an interesting attempt by Józef Maria Hoëne-Wronski to create a purely numerical definition of the mathematical constant π which is independent of the classical, geometric definition of “the ratio of the circumference of a circle to its diameter.” This has been a goal of many mathematicians, since the idea of π seems like it is more fundamental to mathematics than a definition based on circles would make it seem– as evidenced by the fact that it shows up in areas of mathematics which are seemingly unrelated to circles. Wronski’s idea, to this end, was the following formula:

At first glance, the formula seems inherently nonsensical. After all, is not a number, and therefore cannot be utilized in numerical operations in this way. However, one can get a sense of what Wronski may have *intended* by this equation. It appears that Wronski wanted to utilize to represent an infinite number, and modern mathematics actually gives us several tools for handling this sort of idea. One which might be of particular use, here, is Non-Standard Analysis with its infinite and infinitesimal Hyperreal numbers. In NSA, we have the ability to perform calculations with and upon infinite numbers perfectly consistently and reasonably.

First things first, let’s translate Wronski’s equation into a more modern form. Borrowing from the work Joseph Nebus already did in his second post on the subject, we can replace all the ‘s in the equation with ‘s, instead, in order to get:

Now, we can use our tools from NSA to find suitable substitutes for in the above equation. One immediate problem which a mathematician might notice is that replacing the three symbols with positive, infinite Hyperreal numbers , , and will lead to different solutions for the equation when one uses different values for , , and .

However, Wronski died well before Georg Cantor‘s brilliant work showing that there are different sizes of infinite sets was even published, let alone accepted by mainstream mathematicians. As such, it is very reasonable to assume that Wronski believed his symbol was referring to a single, specific quantity, rather than a range of possible infinite quantities. So, let’s replace all symbols with a single positive, infinite Hyperreal number, . This gives us:

Starting with the expression within the braces, we can explore to find something which may be a bit easier to work with. This takes a little bit of work, but we can show that:

Let’s zoom in on our equation a little bit more, now. The expression is a Complex Hyperreal number which is infinitely close to the Real number, . As such, its reciprocal is also infinitely close to . Given this information, we know that the expression must simplify into some infinitesimal Complex Hyperreal number. Let’s call this number for Hyperreals and .

Similarly, we know that is a non-Complex Hyperreal number which is infinitely close to . Let’s call this number , where epsilon is some non-zero infinitesimal. Multiplying this by our earlier result yields . We can then take this expression and substitute it for the entire braced expression from our full equation:

This, in turn, can now be simplified to:

We’re still far from anything which clearly resembles the π which we all know and love, but now we are getting to a place where we can really start to see some of the implications of Wronski’s definition. Notably, either , or else NSA seems to show that Wronski’s π is not a Real number. As such, it seems like Wronski’s definition is a failure if — presumably, Wronski was not attempting to redefine π out of the set of Real numbers!

However, it seems quite dubious that it would be the case that . Looking back for a moment, we defined our as the Real part of the expression . Let’s break this down a bit further, now. The term is a Complex Hyperreal number which is infinitely close to the Real number, ; let’s call it , for infinitesimal Hyperreals and . I’ll spare my readers a few more convoluted formulae (feel free to work this out yourself!), but if and only if . However, it seems fairly clear that

One of the properties of for all Real is that it has a magnitude equal to 1. This means that for any Complex number such that and are Real and that , it will be true that . The Transfer Principle of the Hyperreals allows us to extend this statement over the Hyperreal numbers, as well. Since the Complex Hyperreal which we were concerned with is , we therefore know that . Since is non-Complex, we know that its square must be positive or zero. Similarly, must be positive and greater-than-or-equal-to 1. As such, the only way for to be true is in the case that .

For this to be the case, then . In order for this to be true, it must be the case that . However, this contradicts our initial definition for as a positive, infinite Hyperreal number.

Unfortunately, it seems that Wronski’s attempt to create a non-geometric definition for π simply does not work. That said, I’m still very curious about his thought process, here. What led him to this particular formulation, in the first place? Is it, perhaps, possible to salvage his work? Could there be some actual truth hidden underneath the apparent incoherence? It will certainly be fun to unravel this puzzle even further.

]]>One of the common claims which is utilized in arguments for the existence of God is that actual infinities cannot exist, implying that there cannot be an infinite regress of causal events in the history of the universe. If there cannot be such an infinite regress, then there must be some First Cause. Theologians then put forth other arguments attempting to show that this First Cause must be God. Blake Giunta, a Christian apologist, has constructed a very interesting and quite useful website cataloging common lines of argumentation from both sides of the debate (color coded Green for Christian arguments and Red for opposing arguments), along with citations and documentation for those claims, called BeliefMap.org. It does not take very long for a fairly cursory perusal of Belief Map to bring one to this exact claim regarding the actually infinite.

While I disagree with Mr. Giunta on many of his views, I have a great deal of respect for him and I think that his work with Belief Map is absolutely fantastic. He truly does attempt to give an irenic and charitable view to the positions of his opposition, and he does sincerely want to discuss the actual arguments being made, instead of being content to knock down Straw Men. To that end, I would like to help Mr. Giunta add to his encyclopedia of apologetics by addressing the manner in which one might answer the claims about actual infinities.

Under the heading, “**Logically, prior events can’t number to infinity**,” Belief Map separates the discussion into two further, green claims. The first of these is, “**Infinity can’t exist in the real world**,” which is further subdivided into three green categories and four red. Two of the reds are theological in basis, and not of much concern to me, but the other two are mathematical and interesting. Each of these red categories contains a minor rejoinder, so I’ll be addressing them as best I can, as well. The second claim under the “**Logically, prior events can’t number to infinity**” heading is that “**Infinity can’t be formed by adding**.” After discussing all of the “can’t exist” categories, I will then consider this one.

The problem with this argument is that it is not even cogent. Infinity is not a number. There are, classically, two ways in which “infinity” is discussed as a concept in the philosophy of mathematics (Katz, 45-50). The first is the “potential infinity,” which is the idea that an iterative process can be repeated without any apparent bound. In this case, “infinity” is a description of the manner in which a process is carried out, and certainly not a number. The second way in which the concept is discussed is the case of “actual infinity,” which is the idea that a completed set can contain a number of elements which is greater than any Natural number. In this case, “infinity” is not a number, but rather a quality of numbers– a number can be either “finite” or “infinite.” And just as there are a multitude of finite numbers, some of which are greater than others, there are similarly a multitude of infinite numbers, some of which are greater than others (Katz 795; Conway; Robinson).

Numerical operations can only be performed upon numbers. For this reason, the expression “Infinity – Infinity” is entirely incoherent. It is no mathematically different than saying “Red – Red” or “Delicious – Delicious” or “Blake Giunta – Blake Giunta.” These are not mathematical statements, and as such, we cannot draw mathematical conclusions from them.

If the argument is amended to discuss the subtraction of infinite numbers instead of the subtraction of infinity, it loses all weight. There exist systems of mathematics in which infinite numbers can be subtracted from infinite numbers perfectly consistently– for example, on Surreal numbers (Conway) or Hyperreal numbers (Robinson). These do not lead to the purported contradictions espoused by apologists.

Belief Map offers three cases as examples of metaphysically impossible scenarios: an infinite tug of war, an infinite hotel, and an infinite popsicle.

The infinite tug of war is actually just a restatement of the question of subtracting one infinite number from another, which we’ve already discussed. It makes precisely the same mistake as before, treating “infinity” as a number and not recognizing that there are numerous infinite numbers, not all of which are equal. As such, it is therefore easily resolved by proper mathematics.

The infinite hotel illustrates a counter-intuitive property of actual infinities, but it does not illustrate a metaphysical impossibility or a contradiction. The only way one might legitimately claim that this is an absurdity would be to already reject the possibility of actual infinites. However, since this is being utilized as an argument in support of just such a rejection, to do so would simply be fallaciously circular question begging.

The infinite popsicle does present something of an absurdity. I’ll agree that Bernardete’s scenario is metaphysically impossible, but not for the reason which he suggests. Popsicles are composed of atoms. Atoms have a significantly non-infinitesimal volume. As such, one cannot create a popsicle with an infinite number of layers in 4 cubic inches (or any other finite volume) of space. This thought experiment doesn’t work, but not due to any metaphysical absurdity relating to infinity.

This bald assertion is an unfortunate bit of question begging. One of the primary definitions of an actually infinite set is a set which contains a proper subset of equal cardinality (Katz 792-795). Belief Map offers no good reason to accept the claim that proper parts always contain less than wholes. One might as well argue that actual infinities can’t exist because actual infinities can’t exist.

Belief Map cites this as an objection to the claim that actual infinities cannot exist, and it is absolutely correct to do so. For more than 100 years, mathematicians have been developing and utilizing a valid and consistent framework for math which deals perfectly well with actual infinities.

However, Belief Map offers a rejoinder to this: “**But so what?** A concept’s being *logically* possible (free of formal contradictions) doesn’t entail that it is *actually/metaphysically* possible.” Certainly, I agree– though, I must say, I find a little bit of irony in this position being raised here, since I quite often see the exact same sentiment brought up by atheists in regards to God’s possibility.

That said, at best this argument is merely inconclusive. It does not say that actual infinities *aren’t* metaphysically possible, but only that they *might* *not *be. In 1925, David Hilbert addressed this precise line of argumentation:

Also old objections which we supposed long abandoned still reappear in different forms. For example, the following recently appeared: Although it may be possible to introduce a concept without risk, i.e., without getting contradictions, and even though one can prove that its introduction causes no contradictions to arise, still the introduction of the concept is not thereby justified. Is not this exactly the same objection which was once brought against complex-imaginary numbers when it was said: “True, their use doesn’t lead to contradictions. Nevertheless their introduction is unwarranted, for imaginary magnitudes do not exist”? If, apart from proving consistency, the question of the justification of a measure is to have any meaning, it can consist only in ascertaining whether the measure is accompanied by commensurate success. Such success is in fact essential, for in mathematics as elsewhere success is the supreme court to whose decisions everyone submits. (Hilbert)

The mathematics of the infinite *has* been successful– inordinately successful, in fact. It forms the basis upon which mathematics has been securely founded. Given that the previous arguments on Belief Map aren’t very convincing– or even coherent, in cases– I see no reason to think that actual infinities are not metaphysically possible.

This is another good objection to the impossibility of actual infinities. If we are discussing an interval with cardinality of at least , then there are an infinite number of subintervals contained therein (Katz 792). Consider, for example, the mathematical interval from zero to one. There are an infinite number of intervals within this interval– for example, and and and , et cetera, et cetera.

Belief Map’s rejoinder to this is to claim that such intervals can only be *potentially* infinitely divided, and are not actually infinitely divided. However, this seems to very clearly not be the case. All of the subintervals on the interval are entirely coextant with . They exist equally as much as the parent interval does and are not simply potentialities waiting around to be actualized.

Now, perhaps Mr. Giunta might respond that he has already granted that such mathematical intervals may reasonably be consistent, but that he is arguing against *physical* intervals, and that these are only potentially infinitely divisible. However, this seems to be yet another bit of question begging, and is only a reasonable assumption if one already denies the metaphysical possibility of actual infinities. After all, if (for the sake of thought experiment) we adopt the assumption that actual infinities are metaphysically possible, then all of the infinite subintervals of a given, physical interval would be coextant with that interval.

Since adding a finite number to a finite number always results in a finite number, Belief Map argues that an infinite collection cannot be formed by the sequential addition of finite elements. However, this seems to be just another circular attempt to reject actual infinities by rejecting actual infinities. No one is suggesting that adding a finite number to a finite number will yield an infinite number. We are suggesting that adding a finite number to an infinite number yields an infinite number, and similarly that adding an infinite number to a finite number yields an infinite number.

To be fair, Belief Map does note that a possible objection to this claim is that past events may “have *always* been infinite in number.” Of course, if it is the case that past events *are* infinite in number, then it *must* be the case that past events have *always* been infinite in number. This is a necessary consequence of the infinitude of past events. It would therefore seem that the claim that “infinity can’t be formed by adding” is entirely irrelevant to the situation under discussion.

From time immemorial, the infinite has stirred men’s

emotionsmore than any other question. Hardly any otherideahas stimulated the mind so fruitfully. Yet, no otherconceptneedsclarificationmore than it does. (Hilbert)

Belief Map, unfortunately, has some ill-formed views regarding the nature and mathematics of infinity. This is owed, at least in part, to the fact that Mr. Giunta borrows heavily from William Lane Craig’s work in the discussion of this subject. However, as I have discussed before, William Lane Craig has a gross misunderstanding of the concept of infinity (Part 1 and Part 2). Hopefully, the information which I have presented here can help Mr. Giunta to improve his wonderful work and correct some of the misconceptions which Belief Map’s arguments present in regards to the actually infinite.

Conway, J. H. *On numbers and games*. A.K. Peters, 2006.

Hilbert, David. “On the Infinite.” 1925. URL: https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Philosophy/Philosophy.html

Katz, Victor J. *A History of Mathematics: An Introduction*. Pearson, 2018.

Robinson, Abraham. *Non-Standard analysis*. North-Holland Pub., 1974.

In the video, Dr. Wildberger claims that there are three different ways in which is commonly discussed: the Applied, the Algebraic, and the Analytical. He does a fairly good job of discussing the manner in which the ancient Greeks discovered that there exists no ratio of two whole numbers which can be equal to , which is a topic I have covered here, as well. He then explains what he means by each of the above three categories.

Since we have shown that there is no ratio of two whole numbers which can equal exactly, the Applied path seeks to find ratios which simply come close to equaling that number– approximations with an arbitrarily large or small error. We are not searching for an exact solution, on the Applied path, and indeed we are content to agree that there is no exact solution which can be attained, according to Dr. Wildberger. We can, for example, find that 1.414, when squared, gives a solution quite close to 2, but it is not exactly 2.

For the Algebraic path, we can construct an extension to the rational numbers which contains some exact solution to the question of — Dr. Wildberger gives the example of an arithmetic using pairs of rational numbers and such that . He notes that this can be done in such a way that it conforms to all the usual laws of arithmetic, but objects that the in this scenario “has nothing whatsoever to do with that one-point-four-one-four-et-cetera that we were talking about previously.”

Finally, Dr. Wildberger presents the Analytic path, which he describes as “the square root of 2 is some infinite decimal which starts out 1.414 and goes on in some fashion.” He unequivocally refers to the Analytic path as “wrong thinking,” and unabashedly goes on to claim that such an object “does not exist, my friends.” It is quite clear that Dr. Wildberger has no love for Analysis. Quite the contrary, he is openly hostile to the idea.

While there are minor statements that I could nitpick in Dr. Wildberger’s treatments of the Applied and Algebraic approaches to the square root of 2, it is his handling of the Analytic approach with which I’ll interact in this article. His discussion of the subject is incredibly hyperbolic, highly oversimplified, and entirely uncharitable. Dr. Wildberger doesn’t even pretend to consider the idea that the Analytic approach may have some reasonable underpinnings which he nevertheless finds to be flawed; rather, he simply dismisses the entire field of analysis as being incorrect and accuses it of being the ruin of mathematics. He treats the subject in this manner despite the fact that, as he will well admit, the overwhelmingly vast majority of all the world’s mathematicians from the past hundred years find the Analytic approach to be perfectly good. In fact, Dr. Wildberger rather boldly claims that these other mathematicians “are all wrong. They are seriously wrong.”

The closest which Dr. Wildberger comes to giving an accurate description of the Analytic approach is when he is discussing the number line. According to Dr. Wildberger,

…this Analytic approach to root 2… pretends that, somewhere on the line (which up ’til now only consists of Rational numbers), somewhere there’s a new place, and it’s somewhere between 1 and 2, and there’s a new number called ‘root 2,’ and it has the property that its square is 2, and we can find out what this thing is by making a calculation.

To say that this is a mischaracterization of Analysis is quite an understatement. In truth, Analysis is based upon an assumption regarding the number line, but it does not simply try to plop an object called somewhere between 1 and 2, as Dr. Wildberger claims. Rather, the assumption regarding the number line upon which Analysis is built is a fairly reasonable one– the idea that the number line is continuous. That is to say, Analysis assumes that there are no gaps or holes in the number line. If the number line only consisted of Rational numbers, as Dr. Wildberger claims it did, then there would be a great many holes in it, indeed, as there are a great many mathematical statements which produce values which cannot be expressed as Rational numbers– uncountably infinitely many, in fact.

The idea that the number line is continuous did not originate with Analysis. It had been an openly discussed question in mathematics since at least the ancient Greeks. The Analysts simply decided to explore what it would mean for such a continuum to exist. Quite happily, they found that assuming continuity led to very beautiful developments in mathematics– exactly the opposite of the picture Dr. Wildberger paints.

If one assumes that the number line is continuous, as the Analysts did, then there is no need to try to create a place for to go, despite Dr. Wildberger’s intimations otherwise. It’s already there, occupying a gap between the Rational numbers. Analysis simply asks, “What can we learn about this gap?” It was not arbitrarily placed between 1 and 2, as Dr. Wildberger hints. Analysis helps us to discover that it is there.

Nor is it true that Analysis claims “we can find out what this thing is by making a calculation.” We already know what this thing is: it is the square root of two. Dr. Wildberger is conflating “what this thing is” with the manner by which we symbolize this thing when using a particular notation. That is to say, Dr. Wildberger is attempting to claim that the number **is** its decimal representation. This is why he takes such offense at the ellipsis which is used to show that the decimal representation is incomplete. For Dr. Wildberger, the decimal representation **is** the number.

This is, of course, a silly notion. The symbols which we use to represent an idea are not equivalent to that idea. Nobody thinks, for example, that the color blue necessarily consists of the letters “b,” “l,” “u,” and “e.” Nor would anyone claim that is a more proper symbol for the number it represents than is “two” or “два” or “二” or || …or even . Similarly, it seems more than a little misguided that Dr. Wildberger is so inordinately attached to the decimal representation of the square root of 2. The fact of the matter is that, so long as it is clear that we are talking about the square root of 2, then it doesn’t matter if we represent that notion with or with or with or with “the ratio of the magnitude of the diagonal of a square to that of one of its sides.”

So when Dr. Wildberger writes…

…and asks, “Is this a correct and meaningful statement?” the answer to both is, “Yes.” None of the displayed digits is incorrect and the ellipsis acknowledges that the display is incomplete. This statement gives us a good bit of information about , and that alone makes it meaningful. When Dr. Wildberger asks about moving the ellipsis to display fewer and fewer digits, the expression remains correct and meaningful, but becomes less useful as we omit more information. The simple fact of the matter is that a mathematical statement can most certainly be “meaningful” without carrying perfectly complete information.

Even when Dr. Wildberger presents the question of in an attempt to show that the ellipsis is absurd, he is misguided, as this statement actually does have meaning– it tells us that is equal to a number. Now, Dr. Wildberger is correct to point out that one is not likely to get any credit for such an answer on homework or an exam, but his reasoning is incorrect. As Dr. Wildberger well knows, good math homework and exams care less about the completeness of the answer than they do about how the student arrived at that answer. After all, which should receive more credit on a test: a correct answer with incorrect work shown or an incorrect answer with the correct work shown? So, while may be a *technically* correct response, it does nothing to show that the student has any understanding of whatever mathematical concept is actually being tested.

This idea that the decimal expansion of contains an infinite number of non-repeating digits seems to be the only real objection which Dr. Wildberger presents in this video, but his opposition to it seems misplaced, at best. In the description to the video, Dr. Wildberger notes that he will further discuss the logical problems which he purports to exist in the treatment of irrational numbers in his videos on Cauchy sequences and Dedekind cuts, so I will be sure to watch these as well; however, his bold pronouncement that “none of them work” seems more than a little arrogant. We’re not talking about some fringe development in a little known field which is sparking controversy and debate. On the contrary, Dr. Wildberger is overtly stating that hundreds of years worth of the world’s greatest mathematical discoveries are completely wrong.

I believe I understand why Dr. Wildberger makes such outlandish claims. In some of his other work, I have seen him explicitly reject the axioms of infinity and of choice utilized in modern set theoretic frameworks. Certainly, without these axioms, our understanding of the irrationals becomes far less rigorous. However, Dr. Wildberger’s aversion to these axioms has led him to caricature his opposition rather than to treat the opposing viewpoint with even the remotest sense of charity. As such, it seems fairly difficult to take his claims on the subject seriously.

Norman Wildberger’s video on the square root of 2 does not contain the “inconvenient truths” which it purports to show. Worse, it contains rather convenient falsehoods which Dr. Wildberger has utilized in his attempt to denigrate Analysis.

]]>We can use a function for just such a purpose. A function is a specific mathematical tool which allows us to describe an entire set of data points all at once which we symbolize as (read “ of “). We encode the data by means of a mathematical formula. For example, our exemplary rolling ball might well have been encoded by the function , where the represents the time, in seconds, that the ball has been rolling, and the value of the function, tells us the distance in meters which the ball has traveled in that time. In this particular function, the coefficient of tells us the rate at which distance changes as time passes– that is, a meter per second. When the boy first rolls it, the ball is traveling at a meter per second; when it finishes it had been traveling at a meter per second; and at any single point during the journey the ball is traveling at a meter per second.

However, this is a very simple example. It describes a situation involving a constant velocity. Things become a bit more muddied when the rate at which a change occurs is, itself, changing.

Our example above describes a **linear function**. Linear functions are so named because they can be graphed on a Cartesian plane to form a straight line. The equation for a linear function is of the form , where represents the y-intercept (the point at which the line crosses over the y-axis of the plane) and where represents the slope of the line (the rate of change for the function). Utilizing the function from our example, , we have a slope of , an intercept of 0, and we can produce the following graph:

It’s very easy to see, intuitively, that this line’s slope, or rate of change, is constant throughout the whole function. We don’t even need to see the equation which generated this graph to see that this is the case, if we presume that the line on the graph is actually as straight as it appears. That very straightness is precisely what we mean by a constant rate of change. As such, it is perfectly clear that the graph has the same slope at as it does at or or . Regardless of how far along the graph we look, it will always have the same rate of change.

However, this is not true of all graphs. When a function ceases to be linear, the rate of change of that function ceases to be constant. Take, for example, the following graph of the function :

Let’s pretend that, instead of rolling the ball across a flat floor, the little boy has instead set the ball atop a ramp and let go. The ball starts moving slowly, but builds up more and more speed as it moves farther and farther from the boy. After four seconds, the ball is two meters away from the boy– just as in our first example– which means that the ball still traveled a meter per second, overall. However, it seems entirely clear that the ball was not moving at that speed at every single moment in the journey, the way it had when the boy rolled it across the floor. At the start of its roll, the ball is moving much slower than a meter per second, while at the end it is moving much faster than a meter per second.

This introduces a very interesting, and very important, question: how can we tell what the rate of change is at any given point? What is the **instantaneous rate of change**?

For example, let’s say I want to know how fast the ball is moving precisely 3 seconds after the boy has set it rolling. A person might think that they can simply determine how far the ball has gone in that time– meters– and then divide that distance by the time– 3 seconds– to conclude that the ball is traveling at of a meter per second. However, this has the same problem as the whole 4 second journey: the ball seems to be traveling slower than of a meter per second at the start and faster than of a meter per second toward the end.

One way in which we know this fact is by looking at how far the ball travels between the second and third seconds of its journey. So, after two seconds, the ball is a meter from its starting point. After three seconds, it is of a meter from the starting point. This indicates that the ball traveled of a meter in one second. But this, again, falls prey to the same problem we’ve been having: the ball seems to be moving more slowly than of a meter per second at the 2 second mark and more swiftly at the 3 second mark. We’re closer to the speed of the ball at 3 seconds than we were before, but we still haven’t determined it, quite yet.

We can continue to take smaller intervals of time in order to find better and better approximations of the speed of the ball at the 3 second mark. For example, using the distance the ball moves between the 2.5 second and 3 second marks, or the 2.75 second and 3 second marks, or the 2.99999999998 second and 3 second marks. We can come really, really close to the answer we’re trying to find by doing this, but we don’t end up with the exact answer– and mathematicians are not happy to settle for an inexact answer.

Let’s think about what we are doing in these approximations.

If the ball had traveled at a constant speed from the start, at time 0, to the 3 second mark, then its journey could be represented with line . The slope of this line is — which is the approximate speed we determined when considering the ball over this period. Similarly, the line has a slope of , our approximation from the 2 second mark to the 3 second mark. If we were to calculate the slope of , our approximation would get even better. Visually, in the graph above, we can see that the linear graphs are getting closer and closer to the parabolic graph– but there’s always some tiny bit of space between the two.

Algebraically speaking, what are we doing in these approximations? How can we translate this problem into our mathematical language?

Well, we are taking the distance which the ball has traveled after 3 seconds– which, in our math language, is — and we are subtracting the distance which the ball had traveled at an earlier time– say, or or — to find the distance which the ball has traveled between those two times. We are then dividing this distance by the amount of time which has elapsed between the two points: seconds or second or seconds.

Now let’s try to generalize this. We have our function, . We are looking at the difference between the value of the function at some point, , and the value of the function at some subsequent point, ; we are then dividing that difference by the difference in our two points, — which is just . So, this leads us to the expression .

As we have seen, the smaller the gap between our two -values, the closer our approximation becomes. Naturally, we might then think that we can find an exact solution to our problem if we just remove the gap, entirely– that is to say, what happens if we set equal to zero in the expression that we found, above? However, we very quickly come to a problem if we do that. Evaluating the expression, we’ll see that . This is certainly problematic– any middle school child should be able to point out that we simply cannot evaluate that .

But what if we had some number which wasn’t zero, and yet that number was infinitely close to ? In such a case, we could reasonably assume that our answer is infinitely close to being correct.

Thankfully, in the first part of this series, we learned that we do have such numbers: the infinitesimals. So now, if I replace the from our above expression with any arbitrary infinitesimal– let’s call it — we’ll find that evaluates to something infinitely close to the answer which we are looking to find. For the exact answer, as we mentioned, we would like to have been able to replace the with zero; but now we can be clever. Instead of trying to do undefined operations of math, dividing zero by zero, we can find the Real number solution which is infinitely close to our evaluated expression, which (as you will recall) is called the **standard part** of the expression. By taking the standard part of , we can find the exact answer to our problem.

Let’s go back to our rolling ball, now, to see how we can put this into use. We want to find the exact speed of the ball at the 3 second mark. Translating this into our expression, we get:

So, precisely at the 3 second mark, we now know that the ball is traveling at exactly of a meter per second. However, we can do even better than this. As mentioned earlier, mathematicians are greedy. We don’t just want to know what’s going on at a few of the points; we want to know what is going on at *all* of the points. So, rather than solving for some particular value of , such as 3, we can solve the expression for *all* values of , like so:

This new function, , is called the **derivative** of our original function. We denote the derivative of with an apostrophe, written as and often read as “-prime of .”

The derivative is a very powerful tool. It gives us a way of describing the instantaneous rate of change for *all* points of a given function. When discussing speed or velocity, as we have been doing for our exemplary ball, the derivative of the function for distance gives a function describing velocity. The derivative of the function describing velocity will, in turn, give us a function describing acceleration. Taking the derivative of that function will then tell us how quickly our acceleration is, itself, increasing or decreasing– and so on and so forth. When we take derivatives of derivatives, like this, we refer to them as second, third, fourth derivatives (and so on). So, as we have now seen, the second derivative of a distance function is an acceleration function.

The derivative was developed by mathematicians for the express purpose of describing the changes in change. By its use and exploration, we can conquer a great many problems which are incredibly difficult– or even impossible– without this wonderful tool. And, at the very heart of the derivative lie the infinitesimals– these numbers between our numbers– which give this mathematical tool its power.

]]>There are numbers in between the Rational numbers, too. We can define some number, , which is not equal to any Rational number. There are Rational numbers which are greater than , and those which are less than , but somehow our number squeezes itself into a gap in between the Rational numbers. In order to find such a number, we need to further extend our understanding of “number” to include the Real numbers. This should all be very familiar to the average high-school student.

Now, what happens if we extend this idea one step further? Are there more numbers which are in between the Real numbers?

For thousands of years, mathematicians have had heated debates about this question. There is a well-known concept in number theory called the Archimedean property, named after the famous mathematician Archimedes (though he, himself, had attributed the idea to his friend and mentor, Eudoxus). Euclid described the notion by saying, “Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.” In short, this means that given any two numbers, and such that , then we should be able to add to itself a finite number of times in order to find a number which is larger than . For example, given the numbers 5 and 34, I can add 5 to itself seven times in order to get a number greater than 34– that is to say, . Given the numbers and , we can find that . Given the numbers and , we can find that .

However, this property led to a very curious problem when mathematicians began trying to discuss the number of points contained in a given line– particularly when those mathematicians attempted to compare the number of points in one line to the number of points in another line. The eminent philosopher, Aristotle, came to the conclusion that such discussions could be nothing but nonsense, and that any attempt to quantify the number of points in a given line would simply lead to confusion and folly. As an example of this, Aristotle discussed what has come to be known as his Paradox of the Wheel. Take a look at the following figure:

Ancient Greek mathematicians, while studying circles, wanted to find some way to discuss the circumference of the circle in the same way in which they talk about other magnitudes. So, they began “unrolling” circles to create a straight line equal in length to the circumference of the circle. Aristotle noticed that, given a larger and a smaller circle which share a centerpoint, rolling out the wheel to produce a straight line equal in length to the circumference of the larger circle causes the smaller circle to produce an equally long line. But how can this be? The smaller circle obviously has a smaller circumference, but rolling it out at the same rotational rate as the larger circle makes it seem to have an equal circumference to the big one!

Galileo Galilei, two millennia after Aristotle, attempted to resolve this paradox by arguing that there must be gaps in the continuum– that is to say, there must be empty spaces between the points in any line or figure– and that these gaps account for how the smaller circle’s circumference can be stretched to equal that of the larger circle. However, other mathematicians were quick to note that this would run afoul of the Archimedean principle. If such gaps existed, it should be possible to continue to stretch them until they became noticeably large. We should be able to magnify a line until we literally see it rend apart into pieces.

In the latter half of the 17th Century, Gottfried Wilhelm Leibniz began to argue that there are numbers which are infinitely small. To the absolute shock of the mathematical community, Leibniz was claiming that a 2000-year-old immutable law of number theory was, in fact, incorrect. There existed numbers, Leibniz claimed, which violate the Archimedean principle; numbers which are greater than zero, but which are nonetheless so much smaller than any Real number that it is impossible to find a finite ratio between that infinitely small number and any Real. You could add the number to itself a thousand times, a million, a quintillion, a googolplex– even Graham’s number of times– and that number would still remain smaller than any Real number which you could possibly imagine.

Not only did Leibniz believe that such numbers exist, he utilized them in order to create an entirely new method of mathematics: Calculus. However, the idea was so incredibly controversial that even other proponents of Calculus– like Isaac Newton, who independently developed that field of mathematics– railed against Leibniz for his reliance upon such an insane concept. Still, Leibniz’s results were indisputable, and a number of mathematicians joined with him in an attempt to find some rigorous and logical means of discussing these infinitely small numbers. However, after a great deal of failure, other avenues began to be explored in order to place Calculus on a rigorous footing. Particularly, the notion of the Limit was put forth, expanded, and eventually made rigorous in the 19th Century by Karl Weierstrass. With a rigorous and logical footing finally established for Calculus, the infinitely small numbers of Leibniz’s devising were abandoned and Calculus classes began being taught based on the idea of the Limit.

Thankfully, this was not the end for our strange and tiny numbers. One-hundred years after Weierstrass, and three-hundred after Leibniz, a model theorist named Abraham Robinson began to attack the problem. He was fascinated by Leibniz, and wanted to gain a better understanding of the mind which had invented the calculus. Robinson’s work led to his development of a new number system: one which did not adhere to the Archimedean principle, but which otherwise behaved in exactly the same manner as did the Real numbers. He called this new system the Hyperreal numbers. Just as the mathematicians had extended the Integers to find the Rationals, and then extended the Rationals to find the Reals, Robinson extended the Real numbers in order to find the Hyperreals.

The Hyperreals contain all of the Real numbers, so any number on the Real line is also on the Hyperreal line. However, the Hyperreal number line also contains two very special types of numbers which are not contained in the Reals. The first of these are Infinite numbers– numbers which have an absolute value greater than that of any Real number. That is to say, we can define some number such that, for any given Real number , it is true that . The second type refers to Infinitesimal numbers. Infinitesimals are the reciprocal of Infinite numbers, and as such, have an absolute value which is smaller than that of any Real number (except 0, which is considered to be Infinitesimal): .

Any number which is not Infinite is called a Finite number– including the Infinitesimals. Once the system is in place, it becomes quite easy to prove some simple, but powerful, properties of the Hyperreals. Given any positive Infinite numbers, and ; any positive Real numbers, and ; and any positive, non-zero Infinitesimals, and ; we can derive the following:

- , , and are Infinite
- and are Infinite
- , , and are Infinite
- is Finite (and possibly Infinitesimal, in the case )
- is Finite and non-Infinitesimal
- and are Finite and non-Infinitesimal
- , , and are Infinitesimal

You may notice that there are several cases missing from the above list. These cases are indeterminate forms– that is to say, without knowing more about the particular numbers involved, it is impossible to tell whether the result will be Infinitesimal or Finite or Infinite. The indeterminate forms are:

We can also derive another very important notion:

For any Finite Hyperreal number, , there is exactly one Real number, , such that is Infinitesimal. In such a case, we call the **standard part** of , denoted as .

Any two numbers which are only separated by an Infinitesimal are said to be **infinitely close** to one another. As such, another way of wording the above is that the standard part of any Finite Hyperreal number is the Real number which is infinitely close to it.

So, now, we can answer the question with which our article started. Are there numbers between the Real numbers? We find that the Hyperreals allow us to answer this with a resounding, “Yes!” Given any Real number, , and any Infinitesimal number, , we can be absolutely certain that there are no Real numbers which come between and . This concept is the absolute foundation of Infinitesimal Calculus.

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