Quick, let me test you: is ‘C’ the first letter that appears on the first page of "Crime and Punishment" by Dostoevsky. Answer A: The first letter is ‘C’. Answer B: The first letter is not ‘C’. Well for those who answered A and for those who answered B, I'd like to ask the same thing: ‘how do you know?’. And before you close this article having felt the philosophical-logical-intuitionistic jargon that is to follow, allow me to tell you that whatever you picked, you picked based on conjecture (=a conclusion formed on the basis of incomplete information, as of google search). In other words, you wouldn’t know of all the instances of “Crime and Punishment” hard-covers so as to naturally infer the quality of the first letter; you answered based on unconstructive proof of the type: “ if it has to be between option A or option B I will choose A (or B) because I can’t disprove my option, or prove the other option”. Notice how this type of proof doesn’t rely on you physically pointing to the first letter in each book of C&P in the observable universe (which would be constructive proof)- it simply infers the truth, on the premise of the excluded middle which sounds like:“the first letter can either be <C> or <not C>” OR “there is proof that the letter is <C> or there is no proof that the letter is <c>”. It is a false dilemma, if you will, that logical fallacy in which you savagely and erroneously don’t include other options apart from the disjunction(=the relation of two distinct alternatives, as of google search).
However! What if it were as simple as saying: “it depends- the first letter is neither <C> nor <not C>- it is unknown.” It is just like saying: “I don’t judge based on conjecture, I judge through direct proof of one of the options” Moreover, it directs to a couple of logical consequences i.) both statements are true at the same time ii.) both statements are false at the same time iii.) both statements are true and false at the same time. Look, the ‘inclusion’ of the excluded middle may not sound like that big of a deal in the context of everyday life such as: my best friend is a sleeper agent (whether or not your friend is a sleeper agent or not is neither true nor false unless they reveal themselves to be one). Even so, it is quite the deal in the context of mathematical existence. One would expect Maths to be rigorously exact, with the numbers existing outwardly of the mind (in platonism) and their manipulation being simply a reflection of the human intellect. Or, you take the anti-realistic view of “numbers are a mental creation whose existence is valid only if verified by constructive proof”. In other words, in mathematical systems that approve of intuitionism, a number's existence is dependent on the number being constructed in the mind with proof; proof is here an instrument of mathematical construction, strictly an activity of the mind. This is merely a show of intuitionistic logic, Brouwer, the head of the intuitionistic movement, defined mathematics as a “languageless creation of the mind”[see article in Stanford Philosophy Encyclopedia]- not a play of symbols that describe real world scenarios, but a stand-alone entity of the mind that exists without the language and that is formulated on the feeling of a ‘move of time’(Kantian a priori time). Mathematical language is therefore just the way by which mathematicians alike translate mental objects to each other.
If you’re confused but got to this point, it means you followed through at least 78% of what I said. Thanks! The point was to spark at least a bit of interest in intuitionism in the philosophy of mathematics. As a translation for the article, we started with the Law of Excluded Middle, saw why it’s not accepted in constructivist points of view, got a quick taste of intuitionistic logic and moved on to a wider view of the philosophy of mathematics. For more than just this introduction, check out these videos from Attic Philosophy (vid1- for understanding constructive proof, vid2 - the philosophy behind intuitionistic logic) and this article that goes through all the key elements of intuitionism in mathematics and Brouwer’s work.
SOURCES:
Iemhoff, Rosalie, "Intuitionism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Summer 2024 Edition), Edward N. Zalta & Uri Nodelman (eds.), forthcoming URL = <https://plato.stanford.edu/archives/sum2024/entries/intuitionism/>.
photo credits: Tommy Parker, UK
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