Boxing Pythagoras

Intuitionism and the Excluded Middle

Introductory lessons on Logic often make note of three basic, but powerful, principles which are so universally recognized that they are commonly referred to as the Laws of Logic. The first is the Law of Identity which states something like, “A thing is equal to itself.” The second is the Law of Non-Contradiction, sometimes phrased as, “A proposition cannot be both true and false at the same time.” The third is known as the Law of the Excluded Middle which declares, “Either a given proposition is true or else its negation is true.”

A classic example of the Law of Identity might be, “Socrates is Socrates.” An illustration of Non-Contradiction could be, “Socrates cannot both be mortal and not be mortal at the same time.” For the Excluded Middle, we would say, “Either Socrates is mortal or else Socrates is not mortal.” This all seems perfectly obvious and simple, even to complete beginners in the study of Logic.

However, one might be surprised to learn that the Law of Excluded Middle is actually a source of some controversy in philosophy– particularly in the Philosophy of Mathematics, where there exists a small but strong community which rejects this principle vehemently.

The Law of Excluded Middle has been an extremely useful tool in mathematics for millennia. It is the basis for a very powerful, and very common, method of proof called Proof by Contradiction. The idea of a Proof by Contradiction is that, given any proposition , the Law of Excluded Middle tells us that either is true or else (read as “not “) is true. With that knowledge, we can presume and start working through some implications of that proposition. If that assumption leads us to assert a contradiction, then our assumption must be false. Finally, if is false, then once again we know that must be true.

For example, let’s consider the proposition, “There exists some positive rational number, , such that is less than all other rational numbers.” Let’s presume that this proposition is true. We can then examine another positive rational number, such as . However, this introduces a problem: since we have contradicted our earlier assumption that is less than all other positive rational numbers. That assumption must therefore be false and we have now proved by contradiction that there is no positive rational number which is less than all other positive rational numbers.

To sum this all up succinctly with symbolic logic, we traditionally affirm that , known as Double Negation, which is to say “not not- implies . While double negatives are typically frowned upon by English teachers they are very commonplace in Logic lessons. For example, if I told you that my apple is not not red, you would likely conclude that my apple is red.

This all seems perfectly reasonable and sound. However, in the late 19th and early 20th Centuries, a different brand of the philosophy of mathematics began to take shape. It came to be known as Intuitionism and it had a very interesting and peculiar take on this whole idea. Intuitionism rejected classical logic, in which all statements must either be true or false. To the intuitionist, this notion was short sighted because he asserted that there exists an entire class of statements which cannot rightly be considered either true or false– nonsensical or meaningless statements. For example, if I were to make the statement “Blue times Sonorous equals John Wilkes Boothe,” and asked you whether it was true or false, you would be rightly confused. The statement doesn’t even make sense! How can we assign a truth value to it?

The Intuitionists created a new type of logic which was epistemically grounded. Rather than discussing what must-be-true and what must-be-false, Intuitionist logic dealt with whether or not a statement is provable. The difference is subtle, but very important. If I am given the proposition , then the statement , on Intuitionist logic, means “one cannot prove .” Following this down the rabbit trail, the statement then means “one cannot prove that one cannot prove .” This is extremely important because the fact that one is unable to prove that one can’t prove does not imply that one therefore can prove . That is to say that , our Double Negation theorem from classical logic, is not a tool which we can use in Intuitionist Logic.

Here’s an example– let’s imagine that we have a large glass jar filled completely with sand from the beach. The jar is completely sealed, and we have no way of opening it in order to count the grains of sand. Now, we know for a fact that either the number of grains of sand in that jar must be even or else that number must be odd. Since we cannot count the grains, we cannot prove that the number of grains is odd. However, the fact that we cannot prove the number of grains is odd does not imply that we can prove the number of grains is even. Despite the fact that the number must be odd or else even, neither proposition is provable given the rules of this scenario.

Because of this focus on epistemic provability, the Law of Excluded Middle does not exist in Intuitionist logic. Our earlier proof that there is no least positive rational number would be invalid on the Intuitionist system, and a much stronger method is required in order to effect the same proof.

This may all seem rather pedantic, to the casual reader, but there are some very important, far-reaching implications when it comes to mathematics. It is no accident that Intuitionism arose at the time in which it did. It was in that same period of time that the mathematics of infinity began to jump by leaps and bounds, solving numerous long-standing questions of math but simultaneously introducing just as many extremely difficult and curious new ideas which were at best counter-intuitive and at worst downright paradoxical. A large portion of the foundations of this new infinite mathematics had been built upon Proofs by Contradiction, Double Negation, and the Law of the Excluded Middle. The Intuitionist rejection of this classic logical law and the subsequent formulation of a whole new system of logic was a direct response to these claims about infinity in mathematics.

At the time, Intuitionist philosophy was very strong in the philosophy of mathematics and by the 1920’s the community was fairly split on the subject. However, the Law of the Excluded Middle and the infinite mathematics which it enabled proved to be such enormously strong tools that Intuitionism was eventually overpowered. One of the most influential mathematicians of the 20th Century, David Hilbert, once responded to the Intuitionists by saying, “Taking the Principle of the Excluded Middle from the mathematician … is the same as … prohibiting the boxer the use of his fists.” By and large, the mathematical community began to agree with this position more and more; however, Intuitionism never died out, entirely. Even today, there remains a small but vocal minority within the philosophy of mathematics fighting for the Intuitionist system.

Anyone who has been reading this blog for some time will know that I am absolutely enthralled with the mathematics of the infinite. When I first encountered opposition to this field of math, I was shocked. When I then learned that this opposition rejects such a basic and universal principle of logic as the Law of the Excluded Middle, I became downright dismissive. However, the more I’ve read about Intuitionist philosophy, the more I’ve come to respect it– especially its most famous proponent, L.E.J. Brouwer. Of course, I still don’t agree with the Intuitionists, and I still find the mathematics of the infinite to be a beautiful and wonderful world of philosophy; but I do have a much better appreciation for the philosophical footing of those with whom I disagree.