'Proof' Review: Finding Truth in Numbers
wsj.com36 points by Hooke 13 days ago
36 points by Hooke 13 days ago
I don't like how they give and example of a geometric axiom and then give a number theory result. This makes it seem that the number theory result follows from the geometric axions.
> I don't like how they give and example of a geometric axiom and then give a number theory result. This makes it seem that the number theory result follows from the geometric axions.
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.
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"If the axioms are true, and the subsequent reasoning is sound, then the conclusion is irrefutable. What we now have is a proof: something we can know for sure."
... 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.
Well, in modern mathematics we don't presume that the axioms are 'true' in any meaningful sense. All of modern mathematics is conditional.
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.)
When you accept axioms randomly and reject even logical semantics, you wind up working with something like the so-called rado graph of Erdos. Call the first set of assumptions/structure meaningless and choose a new one, also having no grounding in reason - it turns out that would make no difference. "Almost certainly" you wind up with the same structure (the rado graph) in any case.
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.
I am accepting and rejecting axioms _arbitrarily_ depending on application, but not _randomly_, and definitely not i.i.d. random.
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.
Rado has many interesting subgraphs, for example the implication graph of every countable theory. Are arithmetic propositions associated randomly? Not as we see it, but in some sense they could be (even iid). It is not the quite the same but I take the rado construction to be the logical extreme of the position there is no inherent/meaningful truth in such theories. You need to deal with this kind of many-worlds absurdity that falls out.
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.
There is philosophy of mathematics. The below are my ideas of my philosophy of mathematics.
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".)
Agreed.
> 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.
> All of modern mathematics is conditional.
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).
> 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.
Yes, the law of the excluded middle is an interesting one to exclude. You could also try and go without certain infinities: eg you could remove induction (and things equivalent to induction) from your toolset, and see how far you can go.
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.
> All of modern mathematics is conditional.
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.
Which axioms you take as true is a free choice. They aren't true or false by themselves.
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.
you cannot prove the consistency of a system of proof within that system, ie at all.
There are systems that we can proof the consistency of just fine. See eg https://en.wikipedia.org/wiki/Presburger_arithmetic
Presburger Arithmetic cannot prove its own consistency inside of itself, but that doesn't mean we can't prove its consistency 'at all'.
What on earth do you mean by prove, If not within a system of proof?
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.
But it's not really relevant for anything to be able to prove that you cannot prove the inconsistency.
But proving inconsistency can be done - show that a contradiction follows from the axioms.
"What's irrefutably proven is that if you take this particular set of axioms, then these conclusions hold."
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.
We do end up with something we know for sure: the whole proposition "if we take these axioms as true, then these statements hold."
As others have noted, axioms are more akin to a line in the sand—they are either so "obvious" as to be true or constitute such a useful and economic basis for further development that we decide to use them.
Mathematics is more about coordinating human observation and discussion than it is about capital T truth (unless you are a Platonist)
Even if you are a platonist, the conditional nature of axioms is still useful.
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.
Axioms and postulates as Euclid uses that term are the same thing. In modern times we have gotten rid of the idea of statements that are self evidently true. So we don’t use the term postulate. We’d call Euclid’s postulates axioms today.
Though to be honest, by modern standards Euclid sneakily uses some extra assumptions in his proofs that you actually need to add as axioms.
See https://math.stackexchange.com/q/1901133/1051561 for some examples.
> We still don't know for sure absolutely.
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.
> we can't know anything absolutely (except that we exist yada yada).
Well we only know that if we 'think'.
Mathematics diverges from reality with infinites... That is where the trouble with axioms starts.
You can do plenty of finite mathematics just fine.
Infinities only play a role in some parts of mathematics.
I dare you to formulate a version of quantum mechanics without it.
Quantum mechanics makes very good predictions.
That talks more to how humans think than to what nature is. It is almost a philosophical debate but I feel gut instinct infinities are not really in nature.
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)
Pi very definitely does not encode infinite information. As you notice, we can write a very short computer program that will produce all the digits of pi (eventually).
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.
I thought it encoded all information but the catch is the index into it contains as much information as what you are trying to get out so it is effectively a very slow, negatively efficient zip format.
That's basically equivalent to saying that the natural numbers encode all information [0], it's just that the index to say which natural number you mean is rather long.
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.
That's right. But it does give a feel for what infinity is to our finite minds. It ain't just big numbers. It's all computation! (For a certain definition of computation)
Enumerating the natural numbers is only a single trivial computation. Enumerating the digits of Pi is a bit more involved, but it ain't universal, either.
Infinity is fully real, in fact it’s the very nature of the finite to pass over into infinity
Not a mathematician but my understanding is that the axioms aren't some universal truth that we discovered, but rather the foundation of our language of mathematics.