'Proof' Review: Finding Truth in Numbers
wsj.com
wsj.com
This isn't some weird gloss on their part; there are number-theory results in Euclid's Elements, even if you and I would nowadays think of them as belonging to a different discipline.
... if the axioms are true. We still don't know for sure absolutely.
"The idea of self-evident truths goes all the way back to Euclid’s “Elements” (ca. 300 B.C.), which depends on a handful of axioms—things that must be granted true at the outset, such as that one can draw a straight line between any two points on a plane."
Strictly speaking, Euclid does not state axioms. He starts with 23 Definitions, 5 Postulates and 5 Common Notions. Drawing a straight line from any point to any point is stated as Postulate 1.
I realize this is a newspaper article.
So _if_ you find a system where the axioms of eg group theory hold, you can apply the findings of group theory. That doesn't make any statement about whether the axioms of group theory are 'true' in any absolute sense.
They hold well enough for eg the Rubik's cube, that you can use them there. But that's just a statement about a particular mental model we have of the Rubik's cube, and it only captures certain aspects of that toy, but not others. (Eg the model doesn't tell you what happens when you take the cube apart or hit it with a hammer or drop it from a height. It only tells you some properties of chaining together 'normal' moves.)
I am not sure this is true for "all" mathematics. You're still using some metalogical axioms that are always true. In particular, laws of untyped lambda calculus (or any other model of universal computation) are "axioms" that you consider unquestionably true (just like you can consider unquestionable that objective shared mathematical reality exists).
Although, if you informally asked me if the rules of logic were true, then I would say that of course they are, if you asked me formally I think I would say that they are not unquestionably true, only unquestionable if you want to do classical mathematics. If you're willing to grant basic rules of logic, then certain consequences follow. If you're not, then you're not doing classical mathematics, although you might still be doing interesting mathematics—for example, if you decide not to accept the law of the excluded middle.
There's quite a bit of ideology / philosophy about excluding the law of the excluded middle. But even if you set these aside, it turns out that 'constructive logic' without the 'excluded middle' has enormous practical applications in eg computer science.
True, and there is some variation in the axioms. But for the record, pretty much all systems keep the logical rules of AND elimination and OR introduction for example: if A and B are true, then A is true. If A is true, then A or B is true. However, the law of excluded middle is sometimes excluded for constructivist reasons.
https://en.wikipedia.org/wiki/Rado_graph
So you can reject meaning (identifying the axioms with simple labels 1, 2, 3, ...) and truth (interpreting logical relations arbitrarily), but in doing so you ironically restrict mathematics to the study of a single highly constrained system.
I only disagree somewhat though - it is all contingent. We do say something like
IF { group axioms } THEN { group theory }
In the previous century there was an attempt to identify mathematics with formalism, but it defeated itself. We need to see truth in these systems - not individual axioms per se, but in systems composed from them. This is justified by the body of mathematics itself, to the non-formalist, the richness of which (by and large) comes from discoveries predating that movement.
The Rado graph you mentioned does look very interesting as a mathematical object in its own right! But it has approximately nothing to do with what we discussed, exactly because it requires i.i.d. randomness.
> Crucially, we then search for the right set of assumptions, and the conviction of modern science that there is a right set of assumptions is "effective" at least.
Well, multiple sets of assumptions can be 'right' or right enough, or perhaps no set can be. Eg as far as we can tell, no clever sets of axioms will allow us to predict chaotic motion, or predict whether a member of a sufficiently complicated class of computer programs will halt. (However, we can still be clever in other ways, and change the type of questions we are asking, and eg look for the statistical behaviour of ensembles of trajectories etc.)
> In the previous century there was an attempt to identify mathematics with formalism, but it defeated itself.
I say it was extraordinarily successful! It actually managed to answer many of the questions it set out at the start. Of course, many of them were answered in the negative. But just because we can now proof that every formalism has its limits, doesn't mean that informal methods are automatically limitless. Or that there is any metaphysical 'truth'.
The 20th century saw enormous progress in terms of thinking about formalism abstractly (that's the 'self-defeat' you mention, thanks to giants like Gödel and Turing). But it's only in the last few years that we've seen progress in terms of actually formalising big chunks of math that people actually care about outside of the novelty of formalisation itself.
Btw, just to be clear: IMHO formalisation is just one of the tools that mathematicians have in their arsenal. I don't think it's somehow fundamental, especially not in practice.
Often when you develop a new field of mathematics, you investigate lots of interesting and connected problems; and only once you have a good feel for the lay of the land, do you then look for elegant theoretical foundations and definitions and axioms.
Formalisation typically comes last, not first.
You try and pick the formal structure that allows you to make the kind of conclusions you already have in mind and have investigated informally. If the sensible axioms you picked clash with a theorem you have already established, it's just as likely that you rework your axiom as that you rework your proof or theorem.
That bidirectional approach is sometimes a bit hard to see, because textbooks are usually written to start from the axioms and definitions. They present a sanitised view.
From your reply though I feel we are not really disagreeing so much. There is a kind of truth which is not propositional or self-evident but teleological. That is a sense in which I think the assigned truth values are meaningful.
The formalist movement was indispensable, I shouldn't have implied it was merely self-defeating. But I believe the philosophy that mathematics is fundamentally arbitrary mechanical symbol manipulation is wrong.
The axioms are true within the system that has those axioms. Therefore, the theorems that result from those axioms and rules, are also true in any system that has those axioms and rules.
It does not make it 'true' in an absolute sense (since it is 'true' within the system and any others (including the Platonic realism, and others too) that includes them), but absolute Truth is inexpressible (this is my conclusion from my study of mathematics and of philosophy of mathematics, but it applies to other stuff too).
However, you should avoid to be confused by such a thing, since some people apparently are. For example, just because some specific sequence of symbols has some use in some system, does not mean that it is the same in a different system (even if they can be mapped to them, which they often can be). Furthermore, even if "X OR NOT X" is true (regardless of what X is, as long as it is well-formed), that does not mean that either "X" must be true or "NOT X" must be true. And, just because values can be assigned to the symbols of classical (or other kind of) logic, does not make it necessary to assign those or any other values.
(Principia Discordia also has some things about "Psycho-Metaphysics".)
> Furthermore, even if "X OR NOT X" is true (regardless of what X is, as long as it is well-formed), that does not mean that either "X" must be true or "NOT X" must be true.
Independent of manipulating meaningless symbols, there's a whole branch of math called 'constructivism' where people try to find proofs without the 'law of the excluded middle'. Ideology / philosophy aside, the methods developed for this curiously handicapped game are of practical interest in computer science.
If we go down the skeptics' route, we can't know anything absolutely (except that we exist yada yada). But we still have to function in the real world, so we assume the most consistent observations will never change. From those observations, we extract the axioms, on which we build the tower of conclusions.
Quantum mechanics makes very good predictions.
Sure you can draw a good circle and say the ratio is Pi and that number encodes infinite information (albeit at high entropy) but to me that Pi is an algorithm for computing better and better approximations to a perfect circle, an object absent from nature.
I don't know QM but I suspect it is the same. It is our mental model. Using infinities is our concept. To me infinite things are algorithms (i.e. as X tends to infinity... Tends being an important word)
In any case, you can do lots and lots of mathematics with either only finite objects, or if you allow limits as you suggest, you can do almost all of math.
Only a vanishingly small part of math deals with actual infinities in a way that cannot be re-written in terms of limits.
However for natural numbers, the 'decompression' is very fast. Much faster than for Pi.
[0] You can make this precise via Bijective Numeration https://en.wikipedia.org/wiki/Bijective_numeration to handle leading zeroes nicely.
Infinities only play a role in some parts of mathematics.
Well we only know that if we 'think'.
See https://math.stackexchange.com/q/1901133/1051561 for some examples.
Mathematics is more about coordinating human observation and discussion than it is about capital T truth (unless you are a Platonist)
Eg look at group theory. It basically says, if you have a set of elements and some operation on this set that satisfies certain criteria (= the axioms of group theory), then you can draw all these conclusions.
I don't think anyone ever argued about whether the axioms of group theory are 'true' in the abstract, because everyone recognises that it depends on your application. Eg they are satisfied for the operations you can do on a Rubik's cube (especially a Platonic ideal of a Rubik's cube), but they aren't true for moves in Sokoban (even a Platonic ideal of Sokoban) nor Tetris.
More famously, look at Euclidean geometry: even setting aside curved spacetime of general relativity, even the ancients knew that Euclidean geometry isn't 'true' on the surface of a globe.
What's irrefutably proven is that if you take this particular set of axioms, then these conclusions hold.
But you are free to choose other axioms, that will lead to other conclusions.
Some statements people use as axioms are equivalent (you can include one, and then derive the other and vice versa). Some are contradictory: you can include the axiom of choice or the axiom of determinacy, but not both as that will lead to a contradiction and thus an unsound system.
In a sense it's a matter of taste, mathematicians choose a set of axioms that leads to interesting things to think about.
This is what I tried to say in my comment. It's the author who talks about the truth of the axioms. I'm objecting to his claim that we end up with "something we can know for sure". No. Your truth depends on your assumptions.
Presburger Arithmetic cannot prove its own consistency inside of itself, but that doesn't mean we can't prove its consistency 'at all'.
You can prove the consistency of PBA, But you cannot prove that you cannot prove the inconsistency of PBA, because The inability to do both is the definition of consistency in the larger system.