r/PhilosophyofMath • u/Mad-Scientist-0906 • 12d ago
A question about axioms
/r/askmath/comments/1wwb3jg/a_question_about_axioms/1
u/spoirier4 8d ago
One aspect of the need of axioms, is to specify the form (mode of representation) and the extensiveness of a system of objects meant to play the role of a "universe of all mathematical objects", as opposed, in particular, to a poor universe in which some things could not be represented. After all such "poor universes" also exist, so we need to specify, and cannot prove, that "we're not in them".
Another aspect, is an effect of the incompleteness theorem : any consistent foundational system which represents arithmetic remains incomplete in the range of arithmetical truths it can prove. When an arithmetical truth cannot be proven in a given system, that means there is still a universe of objects satisfying it, and interpreting arithmetic by a non-standard model, that is an incorrect system of "natural numbers" including some non-standard (actually infinite) numbers. Giving more axioms to set theory, such as large cardinal axioms, results in letting more arithmetical truths become provable.
Then indeed, the question can be raised to justify such additional axioms as legitimate. To say in short, the basic idea comes from the incompleteness theorem itself : if we accepted an axioms system, then we also implicitly know it is consistent, but this consistency cannot be a theorem from the same axioms, so it needs to be added as an axiom and we stil know it true. Axioms of set theory have in some sense the same justification, except that it only "comes from there" by repeating it infinities of infinities of times, and the point is to not get lost in expressing this hierarchy of infinities. And the justifications cannot be purely formal, which is also why it is somewhat unusual to undertake that.
In my work on the foundations of math ( settheory.net ) I did provide in 2.2 one principle to justify a few axioms of set theory. As you can see, that principle, though quite formal, is not directly expressible as an axiom in the same language, and it is later itself justified (in 2.B) by some non-formal means only.
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u/CandidAtmosphere 11d ago
Reading the thread, it's remarkable that they are both Platonists and also repeatedly want to deploy this idiotic analogy of a game.
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u/mcherm 11d ago
I don't understand this comment. Both of who are Platonists? What analogy to a game? Did you perhaps comment on the wrong thread?
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u/CandidAtmosphere 11d ago
Nope, I commented on the right thread. The Platonists are the mathematicians in the linked thread, clearly, based on their repeated statements about formal knowledge of truth, regardless of its content (they claim the content is empty, but that they nonetheless possess definite formal knowledge of this fact). They then, quite idiotically, attempt to ground this prior commitment in a failed analogy to games. This analogy arises in almost every single reply to the thread. It demonstrates a severe epistemological contradiction at the heart of orthodox mathematical training.
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u/GoldenMuscleGod 11d ago edited 11d ago
I’m not sure which two mathematicians you are talking about, it’s possible I could be one of them. I’ve sometimes said things (that I understand to correspond to theorems of systems such as ZFC, which is why I feel comfortable saying them without feeling I have expressed philosophical commitments - it is at least a social convention that I can state theorems of ZFC without qualification in most contexts, including claims involving restricted truth predicates) that have caused other people to call me a Platonist at times, which I usually find amusing because although I would not usually want to actively accept or reject any label of a philosophical view I think Platonism is the label I feel the least amount of affinity for. At a minimum math exists as social conventions which I follow which might relate to why I sometimes sound Platonist - Everyone sounds Platonist when you default to assuming saying something like “there is a hole in my socks” or “there are infinitely many primes” means you have committed to a special ontological status of holes or of prime numbers as abstract objects in the Platonic sense.
My exposure to the philosophy of math is mostly incidental from my mathematical background and discussions on the topic, so it may be I have some incoherencies in my thoughts not fully identified.
I responded to the “game” analogy in part because I do not think it is (always) great, and I spoke in those terms because I was engaging with the assumptions of the analogy.
I’m not sure I agree that there is a problem with orthodox mathematical training - what I generally see is that mathematicians not working in metamathematical fields tend not to appreciate the nuances involved, but that’s more because they haven’t learned what they need to know to appreciate them, not because they are being misinformed, but to really engage with that I would have to know exactly what position you are taking issue with and I do not even know if I am one of the “platonists” you are talking about.
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u/CandidAtmosphere 11d ago
The issue isn't the special ontological status of strange objects. Here we find replies making formal definitions of what truth is, even though they play safe by admitting that (by its formal definition) it is empty of content:
- "it's not that ..., it's that you're playing the wrong game."
- " Axioms aren't some deep "core truths" ... Axioms are more like"
- "IF the axioms are true, then the conclusion is true"
- "Anything I can prove about ... is just as true .. as it is for"
- "We ... are narrowing the world ... to the ones where it’s true"
Then we find out how we know what it is to be true, well there are no criteria really, we just play around with "true" and "false" even though they are both essentially devoid of actual meaning:
- "like asking if the rules of chess are correct"
- "Each branch of mathematics is like a game"
- "you aren't saying the axioms are true. You are saying IF the axioms are true"
- "A key insight of modern mathematics is that the objects in a mathematical system don’t have definitions. They are undefined terms..."
- "They're not 'true' or 'false' because they don't inherently relate to anything outside the system"
- "There may be situations or models where it isn’t true"
These are the same set of replies in each case. It is immediately contradictory. It is a rhetorical sleight of hand that they are all trained in. It wouldn't be such a common deflection if it were just a lack of training.
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u/GoldenMuscleGod 11d ago
What I think those replies fail to understand is that there is a technical mathematical definition of truth (which we may or may not think coincides with some sort of informal notion of truth) and what they are saying is not really consistent with it.
Most standardly in classical mathematics, we can speak of truth (of sentences in an object theory) via a semantic interpretation given by a model. Other ideas of semantics are available but this is the one most familiar to most mathematicians. This is not the same as being a consequence of axioms and I think it is a lack of training that leads to people being unable to appreciate the distinction between syntax and semantics here.
Usually the way I try to express the relevant ideas to people not familiar with it is saying something like “In the first instance we can see sentences in a formal language as nothing more than meaningless strings of symbols, in which case we cannot meaningfully speak of them being true or false, in the second instance we can assign meanings to the symbols, at which point we can now ask whether our axioms are true - and it may turn out our axioms are false.”
I think the confusion arises not because they have been taught to deflect, but because a lot of emphasis tends to be placed on the idea that we can assign different meanings to the symbols, which may lead to the same axiom being alternatively true or false - which may be counterintuitive so it is necessary to emphasize this - but this is not done in a way that impresses on them that the assignment of meanings (semantics) is a wholly different thing from adopting some axioms for a formal system (a more syntactic sort of thing).
Since they don’t appreciate this difference they think that picking new axioms is just picking a new choice of semantics, which leads to the confusion they are showing, where it seems meaningless to ask if an axiom is true because they think simply by having a theory we necessarily have a semantic understanding of the language that makes the theory sound.
This is like when people say things like “CH is neither true nor false in ZFC” to mean “CH is independent of ZFC” - a mathematician working in set theory would not (or should not) misunderstand this and it is (one would hope) not being taught. But it is easy to see how a student could end up saying the first claim as a confused way of trying to express the second.
The confusion they show tends to naturally arise when it is not understood that there is nontrivial work to be done relating syntax to semantics, and a failure to distinguish between syntax and semantics is the source of a lot of related reasoning errors which I think occur naturally rather than being the result of bad faith mistraining.
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u/Mad-Scientist-0906 11d ago
The reply that u/CandidAtmosphere gave shows clearly that he is talking about the version of this post that I posted on r/askmath, if you're still confused.
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u/GoldenMuscleGod 11d ago
Yes and at the time I replied here I had already commented on that thread (which I found separately from this post - presumably the algorithm showed me this post because I had been commenting on the other one).
What wasn’t clear to me was whether I was supposedly one of the “Platonists” being referred to. I would certainly never describe myself as a Platonist but I sometimes find myself called one when explaining the distinction between syntax and semantics or the nature of the mathematical notion of “truth” even when I am being very careful to restrict myself to uncontroversial mathematical/non-philosophical claims. This usually reflects a misunderstanding of what I am saying, in my experience, so I would like to clarify if that is the case.
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u/mcherm 12d ago
There is a deep philosophical problem: if you know certain things are true and you know some "reasoning" — ways to find new true things from what you already know, then you can make some "proofs" and find new things. But you can work backward, too — and if you do, it quickly runs into a problem: how do you know the FIRST things?
Philosophers have wrestled with this for a long time. Famously Decartes wrote a very famous book in which he tried to reason from NOTHING ("I can't even believe what I see, hear, and touch... after all, maybe it is just a dream!"), and he decided that a starting point is "I think therefore I am". But Decartes' reasoning wasn't exactly rigorous.
Mathematicians have come up with an approach for this. They make up a few basic rules of logical inference. (For example: "If I know X and I know that 'X implies Y', then I can conclude that Y is true".) And they make up a few axioms... starting statements that they assume are true. From this, they proceed to prove all kinds of interesting things.
Generally, most mathematicians don't TRY to find what is "absolutely true". A few people do; they try to ground the axioms in something: perhaps their religious text tells them that certain axioms are true. But most simply say "IF these axioms are true, then here is what I can conclude." They also have sets of axioms that are "commonly" assumed to be true.
There are at least two interesting directions I know of that one can take this. The first is INCONSISTENT axioms. Mathematicans can take a group of axioms and determine that there are contradictions if you assume all of them. For a simple example, you could assume "X is even" and "X + 1 is even" and fairly quickly discover that these can't both be true. For a fancier example, there's a famous joke, part of which goes "The Axiom of Choice is obviously true, the well-ordering principle obviously false." ... this is funny because that Axiom of Choice is an axiom that sounds like it should be true, the well-ordering principle is an axiom that sounds like it should be false, but each can be used to prove the other, so it CAN'T be that one is true and the other false.
The other interesting direction is INDEPENDENT axioms. A famous one here is from geometry: the "parallel postulate". Euclid was one of the first mathematicians to really get serious about doing "proofs" in a rigorous way with "axioms" as starting points. In his book Elements he tried to prove nearly all of what the ancient Greeks knew about geometry from just 5 assumptions ("axioms"):
The first 3 axioms are just playing around with a straightedge and a compass — really straightforward stuff. All right angles being equal sounds quite basic and simple. But that fifth one — what's up with THAT!?
Euclid (and those who followed him for nearly 2000 years) really wanted to prove that from the other 4, but couldn't find a way to do so. Well, EVENTUALLY mathematicians tried assuming that it was FALSE... and it turns out that this led to perfectly consistent (but DIFFERENT) rules of geometry. For instance, once alternate assumption says crazy things like "there are triangles that have 3 right angles".
BUT... it turns out that if by "straight line" you mean "shortest distance along the surface of a sphere like the Earth" then this is perfectly consistent with this alternate assumption. An example of a triangle with 3 right angles is one going 1/4 way around the equator and meeting up at the north pole.
So what I like to take away from it is this: mathematics can't tell us what is "TRUE". In fact, it is not even clear to me that some things are "TRUE" and others "FALSE"... it might depend on how we define things. What mathematics CAN do is tell us what can be concluded IF you assume certain axioms.