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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 f

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.

mjw1007 3y ago
I think the Stanford Encyclopedia of Philosophy article is better. https://plato.stanford.edu/entries/intuitionism/ https://plato.stanford.edu/entries/intuitionism/
- barrkel 3y ago
  Yeah, that's much better.
chongli 3y ago
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.
constantcrying 3y ago
>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.
practal 3y ago
> 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?
- constantcrying 3y ago
  >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.
  - denotational 3y ago
    > 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".
bowsamic 3y ago
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.
deleted 3y ago
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mjw1007 3y ago

I think the Stanford Encyclopedia of Philosophy article is better. https://plato.stanford.edu/entries/intuitionism/ https://plato.stanford.edu/entries/intuitionism/

barrkel 3y ago
Yeah, that's much better.

chongli 3y ago

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.

constantcrying 3y ago

>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.

practal 3y ago

> 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?

constantcrying 3y ago
>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.
- denotational 3y ago
  > 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".

bowsamic 3y ago

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.

deleted 3y ago

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