Some of the logical empiricists defended the claim that the basic mathematical axioms are definitions and that the theorems of mathematics are consequently analytic. This position has subsequently fallen out of favor. Critics have alleged that the position was refuted by Gödel’s Incompleteness Theorems, that it is obscure what “analytic” and “definition” are supposed to mean, that there are no “analytic” truths, or that the axioms cannot be definitions. True by Definition revisits this debate. It introduces the distinction between analytic and synthetic truths and explains the controversy around the distinction in the philosophy of mathematics and beyond. The book also defends a radical new position. It is argued that, in certain contexts, the axioms of arithmetic have the status of definitions; however...
I don't want to spoil the suspense. I am of course immediately suspicious of the impulse to introduce the synthetic/analytic distinction.
The author is a guy I know; he and his wife are both quality philosophers. Some of you might be into a work of philosophy of math.
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The Philosophy of Math according to that unknown mathematician, Eric Hines:
The middle step is always "And then a miracle occurs."
Eric Hines
What's at stake in the arguments about arithmetic? I read the whole abstract and chased down some things to understand it, but it's a bit difficult for me to see the point exactly.
It's one of those little things philosophers talk about, in this case whether or not there is truth in arithmetic.
You may recall EN I.3's dictum that, in moral philosophy like ethics and politics, the probably-right is the appropriate standard whereas in math or strict logic one wants a real proof. This is a book about arithmetic proofs, and whether they contain actual truth of some kind. The author will, I believe, stand on the claim that they do, but that the reason that they do is because we are allowed to define some truths which can then serve as grounds for proofs. This includes the axioms of arithmetic, which then can serve as definitions from which truth can be derived in a proof.
The immediate reach for the analytic-synthetic distinction strikes me as the giveaway: https://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction
He then promises an abductive argument for the consistency of our definitions of the axioms. Now, an "abductive" argument is "a form of logical inference that seeks the simplest and most likely conclusion from a set of observations." (https://en.wikipedia.org/wiki/Abductive_reasoning)
In other words, that's the EN standard for practical action in the physical world -- the 'most likely to be true given observations' standard, not a mathematical proof at all. So he's doing exactly what Aristotle warns mathematicians not to do in EN I.3: he's giving a probabilistic account of the proofs.
It's a very interesting topic, although I'm unread in the relevant fields.
As a test, could we say that, in some sense, all arguments begin with some form of defined truth? E.g., the classic deductive syllogism:
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
How do we know all men are mortal, or that Socrates is a man? Aren't those things true by definition? If you say we can empirically observe that all men are mortal, there's the problem of induction. Some unobserved men may be immortal (cf the movie Highlander).
And don't we define what a human being is, ultimately? That may seem odd to say now, but we know in the past there were Neanderthals and other similar species; were they human? Doesn't the answer depend on how we define human?
Without some form of definition that allows us to make connections between things and see them as a kind, all we have is a universe of unique and disconnected things, don't we?
What we say in that case is that the logical form is “valid,” rather than that it is true.
A valid logical form of deduction doesn’t guarantee truth. It guarantees truth preservation. You have to examine the assumptions empirically to find out if they were true. If they were, and the form is valid, the conclusion will also be true. That is called a “sound” argument, a logical argument whose form is valid and whose assumptions are true.
With pure math, there is a problem that there’s not an empirical standard. Definitions as Euclid uses them are just the bare facts of the system. Axioms follow logically from them. The question all these philosophers in this debate are after is: is there a truth behind all this beyond us just stipulating definitions? He is attempting a middle position, but I’m not sure it doesn’t just collapse into another form of definition by raw stipulation. The abductive move is likely just, ‘The best reason to assume that these regular places for drawing the lines relate to something true is that we always keep drawing them in the same place.”
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