The article seems poorly written, or at least self-contradictory in a way that makes me uncertain what it means. The introduction talks about the intuitionism framing mathematics as a human construction, in opposition to an objective reality. But the subsequent paragraph talks about the truth of proofs themselves as being subjective.
It seems to me that these are two different claims. I think mathematics is a human construction and doesn't have a real substance. Instead, mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. But "truth" within a system of assumptions and generative rules is not subjective, it's mechanically provable.
A possible confusion remains in what "true" means. Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality? I think it's trivially true. Gollum was a hobbit, within the system of Middle Earth. If it's not true, then we need a different word for true that does mean this, because this is what most people mean when they use the word about all sorts of imaginary constructs, from institutions to cultural symbols.
> I think mathematics is a human construction and doesn't have a real substance. Instead, mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. But "truth" within a system of assumptions and generative rules is not subjective, it's mechanically provable.
Mathematics is a human construction, but it certainly has a real substance. What does "mechanically provable" even mean, if there is no absolute truth? Do you believe in the definition of a proof or not? Do you believe that whenever the assumptions of your theorem are true, and you have a proof of a conclusion, that then the conclusion is also true? If you do believe that, that's your absolute truth, then. If you don't believe that, a proof is meaningless, isn't it?
I think it's a system of symbols and rules. By mechanically provable, I mean that given axioms (assumptions) and rules, you can devise a machine (i.e. something which follows rules, with no independent thinking or homunculus) which generates statements which follow from the axioms and rules, and this is what "true" means in the system.
So would you say that your system of symbols and rules is real?
Could it be that we both use the same system of symbols and rules, with the same assumptions, but derive different conclusions? If not, why not?
It has no substance other than its representations; there's nothing of it you can touch which is physical. At best, there is a correspondence between the physical and the system.
All information that exists is physical, encoded in a physical substrate. Beads in an abacus, holes in a punched card, distributions of charge in a computer memory. Hypothetical information that is not encoded physically cannot be causal. A book or computer program that have not been written can have no effects. Only information that exists in a physical encoding can be causal, by virtue of it’s physicality.
Not sure where you are going with this. What is causality? Is there a non-mathematical way of making it precise? And if I have found some way to make it precise, does it matter if I write it down here in this HN comment, or on a piece of paper, or just think it? Is the mathematical content different depending on how it is expressed in physical reality?
It does not really matter if the system is sound or not, right? Although of course a sound one is far more interesting. Anyway, any way of justifying this is mathematical (and so would be the definition of soundness, if it was relevant here). If math is not real, then there is no justification.
therefore it is of no consequence if math as itself is "real" or not. it is intended to model whatever "real" even is.
somewhat similarly: in modern logical theories whatever "true" (and/including "false") even mean doesn't matter. is left out of the logical theory and it is effectively a mere parameter.
all the subject does is gurantee "truth in, truth out" (and complementarily "false in, false out")
the precise details of true "and/including" false, seems to me, are somewhere in the boundary between "classical" and "intuitionism" (or "constructivism")
the subtle distinction between intuitionism" and "constructivism" is above my pay grade (and seemingly above the paygrade of everybody I've had the chance of discussing this with)
This is only possible if math itself is real. Note that I am not saying that a particular axiom system like Euclidean geometry has some sort of "real physical manifestation". No, what I am saying is that logical reasoning itself is real. And our reasoning about logical reasoning is certainly real as well, even if logical reasoning itself happens in very abstract form. Math itself might be viewed by some as just a game of symbols. But that doesn't change the fact that the game itself is real. Would it be otherwise, then math would be about as important as chess.
I like to draw a distinction between real and ideal.
I insist that math is ideal. it models reality ideally.
this distinction is important because otherwise we mix together something, and the ideas and concepts (e.g. symbols and rules) we use to describe and model said something.
the game is not real. people playing the game are real, the game getting played is real. the game on its own as may be described in symbols is ideal.
i suppose what it all is all about is the intersection between this reality and this ideality.
You can say that a certain axiom system models a certain part of reality in an ideal way. But whatever is ideal, is also real, because otherwise there is nothing that could model anything. So your intersection of reality and ideality is just ideality itself.
Wow, you excavated an ocean to cover a puddle. how this: "something which follows rules, with no independent thinking or homunculus" can be easier to prove and reason than the initial rules?
For example, what is simpler to reason out: does a chess move violate chess rules, or, there exist a method to construct an electomechanical device, that will Correctly determine whether this chess move is legal?
The reason I brought up a machine explain something as being mechanical is to clarify that it doesn't require intuition.
We can make machines that count. A trivial example: pebbles in a bucket. Neither the pebbles nor the bucket need intelligence to act as (have a correspondence with) a counter.
Completely agree about the machines. Many mathematical results become more rigorous as a result of "can be done on this kind of machine" type of proofs. However, if the goal is getting rid of intuition, machines don't help! Because no matter what the machine (bucket of pebbles or a Buchholz hydra), it takes a lot of intuition to prove, that particular machine correctly enforces intended rules. Usually more intuition, than the original problem.
Without having proof for the machine itself that machine is declared axiomatic. It is a valid way to go about things of course, but I would hesitate calling it "not requiring intuition".
>What does "mechanically provable" even mean, if there is no absolute truth? Do you believe in the definition of a proof or not?
Different Axioms lead to different provable statements.
Believing in standard mathematics basically means that you can not believe in absolute truth. Unless you also believe that some guys a hundred years ago figured the sole and completely perfect rules which totally correspond to reality.
> Believing in standard mathematics basically means that you can not believe in absolute truth.
I agree that if one follows an axiomatic approach strictly and consider "truth" to be a shorthand for "provable from in some logic from some set of non-logical axioms" [1] then one is rejecting any notion absolute truth, since everything is relative to some set of axioms, but I don't agree with the charactedisation of this as "standard"; it seems to me to be a very Formalist stance.
I'd argue that most mathenaticians consider themselves Platonists, and believe that the mathematical objects they are describing are real enough to form some kind of metamathematical "standard model", and "absolute truth" can be defined in the model-theoretic sense relative to this standard model, even if this is somewhat unavoidably handwavy.
[1] : Even if you do think this, "truth" is generally used by logicians in the model-theoretic sense of "truth in some specific model/structure compatible with the language".
>but I don't agree with the charactedisation of this as "standard"
I used it as an objective term, defining mathematical objects on terms of ZFC and truth being relative only to ZFC is the standard mathematical foundation. If you ask a random mathematician what he thinks the foundations of mathematics are it will most likely be ZFC, even if he disagrees with it on any level, it is still what he would set his rival theory against.
Working, or at least claiming to work, in ZFC is fairly standard, but that doesn’t make it the definition of mathematical truth.
As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.
I don’t think it’s contradictory to work in ZFC whilst simultaneously having a non-axiomatic notion of mathematical truth.
I would hazard a guess that most (all?) working research mathematicians would accept the truth of the Gödel sentence for their preferred axiom system (and deductive calculus), be it ZF/ZFC or TG or something else entirely, so I cannot accept the claim that they see the “standard” notion of truth as being relative to all models of some axiom system.
You might think this is just nit-picking, but if it’s fine (in the sense that this is still “standard”) to add an arbitrarily large set of Pi_1 formulae to ZFC from repetitions of Gödel 1, then I don’t think we can say that ZFC is the standard basis of mathematical truth because this cannot be justified in ZFC; there must be some other (standard) notion of mathematical truth used to justify this.
>Working, or at least claiming to work, in ZFC is fairly standard
Which is why I called it "standard".
> I don’t think we can say that ZFC is the standard basis of mathematical truth
You literally just said that working in ZFC is "standard". A standard is a social agreement, standards can be completely false and absurds, while being standards.
>As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.
> truth being relative only to ZFC is the standard mathematical foundation
I think this is at odds with:
> most mathematicians have a sense of truth that is not bound to any axiom system
Re.:
> You literally just said that working in ZFC is "standard"
Nowhere did I say that considering ZFC to be the arbiter of mathematical truth is standard, in fact I’m claiming the opposite.
Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever.
Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”
I don't think so. Mathenatical statements are almost always framed in the context of ZFC. This does not contradict that mathematicians think ZFC is not an absolute truth.
>Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”
All I am saying is that mathematicians are framing their results in the context of ZFC and that this makes it the "standard" theory. I think that statement is absolutely not controversial, even alternative theories are framed in opposition to ZFC.
I absolutely do not think that mathematicians believe that ZFC is "true", as in it is the one and only perfect set of axioms.
My initial argument (and I am sorry if that was unclear) was that IF you believe that mathematical truth is about formal derivations from axioms (ZFC would be such a theory, same as ZF or any variation) then either you have to say that there is one perfect system and all truth is relative to it alone or that there are multiple equally true, but incompatible, theories.
The "IF" is of course important and I don't think many mathematicians actually agree with the IF clause. I actually completely agree with: "Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever." and I am sorry if I wasn't clear. I actually do not think there is any disagreement here.
Yet still professional mathematicians have an underlying notion of truth outside of any axiom systems. I forgot who said it but if we were to find a contradiction using Peano’s axioms, we would say that the axioms were wrong, rather than arithmetic itself.
Even your comment references “perfect rules which totally correspond to reality” which seems to be another way to say “absolute truth”.
>Yet still professional mathematicians have an underlying notion of truth outside of any axiom systems.
I am certain about that, but it does not make my statement less true. I actually think that very few people believe in ZFC as either a formalist absolute or as an arbitrary set of rules. I think the most common view is that it enables other theories, that those mathematicians actually care about. The moment those theories rely directly on axioms things get difficult. I think the following quote describes quite well the state of ZFC:
"The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
But the subsequent paragraph talks about the truth of proofs themselves as being subjective.
This reflects the development of intuitionism. The project was first started by L. E. J. Brouwer, who rejected formalism. It was then developed by his student Arend Heyting, who formalized it with the Heyting algebra, a restricted variant of Boolean algebra that lacks the double negation elimination (~~p => p) and law of the excluded middle (p OR ~p).
Pretty much all work in intuitionistic mathematics continues the work of Heyting. Brouwer would have rejected the entire enterprise as subjective, so he is mainly of historical and philosophical interest.
>mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems.
This is essentially the standard mathematical approach developed during the early last century. From a few basic axioms (which are not really justifiable) new statements are proven and structures are built up. All notions of truth and provability are relative to that system. In standard ZFC (those are the standard axioms) mathematics "1+1=2" is, like all other statements, a statement about sets. The statement is true by the definitions of 1,2, "=" and "+". In an alternative system with different axioms or definitions the statement is false.
This is not the view of intuitionists though. For them the symbols 1,2 or + aren't formalized objects (e.g. sets in this case). They are just symbols, which transfer some (hopefully) shared meaning to another person, who then (potentially with some addition arguments) might also accept the truth of that statement.
In the former the question of "truth" is fully formal there can be no "interpretation" in any meaningful sense. Intuitionism places mathematics fully inside the mind of the mathematician and "truth" can only be found there.
>Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality?
In formalized mathematics "truth" is fully formal. There can not be any "external" truth derived from it, as any such statement is non-sensical.
I agree and this is actually a common problem in philosophical discussions. People struggle to differentiate between two kinds of objectivity: the "total" objectivity of knowledge that is completely context-free and unstructured, and the objective modulo a given set of assumptions (e.g., "the human brain", or at least, the human way of structuring and understanding reality). Mathematical truths are objective with respect to the latter kind of objectivity, but not the former.
It seems to me that these are two different claims. I think mathematics is a human construction and doesn't have a real substance. Instead, mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. But "truth" within a system of assumptions and generative rules is not subjective, it's mechanically provable.
A possible confusion remains in what "true" means. Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality? I think it's trivially true. Gollum was a hobbit, within the system of Middle Earth. If it's not true, then we need a different word for true that does mean this, because this is what most people mean when they use the word about all sorts of imaginary constructs, from institutions to cultural symbols.