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Foundations of Mathematics

the_croftmeister
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7/25/2013 2:33:21 AM
Posted: 3 years ago
As promised, here is a thread on the foundations of mathematics!

I'm not sure how many aspiring mathematicians we have among us (only in the sense that all mathematicians are aspiring to do mathematics) but I thought I would have a crack at it.

Some of the topics of interest that we might discuss:
1. The axiom of constructibility http://en.wikipedia.org...

2. Set theoretic foundations vs. Category theoretic ones http://en.wikipedia.org... and http://en.wikipedia.org...

3. Practical relevance of foundational concerns

4. Constructivism vs. Classicalism (basically concerned with the validity of the law of excluded middle) http://en.wikipedia.org...

5. The consequences of Godel's results on completeness and consistency for first and second order logic. http://en.wikipedia.org...

6. Anything else relevant to foundational concerns!

If there is one particular area that is interesting to a few people, we can always start another thread. I am just gauging interest in the topic since there hasn't been a lot of mathematical discussion in the forums that I've seen.
the_croftmeister
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7/25/2013 2:37:19 AM
Posted: 3 years ago
I might get the ball rolling by asking whether anybody wants to put forward their view on what mathematical entities are. What meaning these numbers and sets and groups and the like have and what their ontological status might be.
Sidewalker
Posts: 3,713
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7/25/2013 4:55:47 PM
Posted: 3 years ago
At 7/25/2013 2:37:19 AM, the_croftmeister wrote:
I might get the ball rolling by asking whether anybody wants to put forward their view on what mathematical entities are. What meaning these numbers and sets and groups and the like have and what their ontological status might be.

That"s the big unresolved question in mathematics, just what is the connection between mathematical truth and physical reality, do mathematical objects exist independently of the human mind or not?
Hilbert and Brouwer said mathematical objects are created by the mind, but on the other side of this, you have Eugene Wigner writing "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" which tells us that there"s got to be more to mathematics than that.

Most mathematicians will admit that they think mathematical objects exist independently of the mind, that we discover mathematical objects and their properties, we don"t invent them. But then the problem becomes, if mathematical objects exist independently of the mind, then how do we learn about them? We can"t have learned it empirically; mathematical objects don"t exist in the real world. Are we born knowing these innate things as Descartes seemed to believe? Are they "revealed" to us by God as Augustine believed? In the end, the problem of the philosophical foundation is an unresolved problem in Mathematics.

Hilbert"s Formalism fell to Godel"s proof that Formalism isn"t possible by showing that it is provable that you cannot prove the truth of all arithmetic statement within arithmetic, and Turing"s demonstration that you can"t have an effective decision procedure. Gottlob Frege"s Logicism fell to the self-referential paradox, even after Russell tweaked it with the Principia, the attempt to reduce mathematics to logic failed. And were were left dangling with Intuitionism which we intuitively feel cannot be true because the challenge to the law of contradiction left us with nothing to logically work with, and it"s foundational postulate that the truth of a mathematical statement is a subjective claim leaves us with an unacceptable vagueness of the intuitionistic notion of truth. In the end, Brouwer's Intuitionism does nothing to explain "The Unreasonable Effectiveness of Mathematics in the Physical Sciences".

Godel showed us that the formalist interpretation of mathematics is not possible, as the logicist wasn"t, and Intuitionism didn"t stand up to scrutiny. There were only really three horses in the race and none of them won.

In the end it became apparent that the greatest mathematical minds ever produced have pretty much collectively concluded that we do not understand the connection between mathematics and reality.

There is one exception, Sidewalker, I do in fact understand the connection, but I"m not telling.
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Poetaster
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7/28/2013 6:56:16 PM
Posted: 3 years ago
I'll first confess that I'm agnostic as to the ontology of mathematical objects and things. However, I'm not devoted to that status, but am willing to defend it. My next confession is that I don't see mathematical statements as being truth-functional. This is because I view axioms as properly modal propositions. For example, I understand the parallel postulate ("Parallel lines never converge") to be a suppressed modal of the form: "It is Euclidean that parallel lines never converge." Because every Euclidean theorem follows analytically from these axioms (that is, the theorems are tautologies of the axioms), the theorems aren't truth-functional either.

Now, if we could program some ideal reasoner with, and only with, Euclid's original postulates, then that ideal reasoner could only believe in theorems of Euclidean geometry. This brings me to what I think mathematics models: ideal reasoners programmed with certain foundational beliefs and rules.

Modeling the physical world using mathematics is something else altogether; mathematically-worded propositions become truth-functional when subjected to that purpose. But describing the physical world isn't what mathematics is is my view, just something that mathematics does.

(I'll break up my original post into two parts for reasons of length.)
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker
Poetaster
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7/28/2013 7:06:16 PM
Posted: 3 years ago
(...continued)
There are many hypotheses which have been shown to be independent of the canonical set-theoretic foundations(Zermelo-Fraenkel set theory: ZF, or ZFC when we add the axiom of choice). There are, however, several proposals to amend these foundations. The axiom of constructability, which the OP mentioned, is one such thing.

Adding the axiom of con. to ZF makes the continuum hypothesis (which is logically independent of the ZF/ZFC axioms), into a theorem. Because I don't feel any pressure to preserve anything ontologically (realism of ZFC, etc.), I don't see the harm in letting the axiom of con. in the door to turn ZFC into ZFCC. It turns the axiom of choice into a theorem and resolves many standing problems in set theory, as well. Sounds pretty good to me. But I'm not a set theoretician, or a mathematician at all, so maybe I'm not in a position to understand their reservations (and it seems like they have many). Maybe the resistance is actually to the prospect of these amendments becoming canonized in some way. But I think this actually would evidence a prevailing aversion to logical pluralism in the mathematical community, making it a cultural/ideological resistance, not a formally justified one.

But I'm mostly speculating here; does anyone want to explain to me why so many set theorists apparently resist the axiom of con. and other amendments to ZFC? Is it more ideological than anything else?

The OP is a real goldmine of talking points, but maybe we should take them one at a time to avoid monster posts so early in the thread!
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker
AlbinoBunny
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7/28/2013 7:19:22 PM
Posted: 3 years ago
I don't know much about this stuff, but sure, I'm reasonably interested.
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the_croftmeister
Posts: 678
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7/28/2013 7:24:26 PM
Posted: 3 years ago
At 7/28/2013 7:06:16 PM, Poetaster wrote:
(...continued)
There are many hypotheses which have been shown to be independent of the canonical set-theoretic foundations(Zermelo-Fraenkel set theory: ZF, or ZFC when we add the axiom of choice). There are, however, several proposals to amend these foundations. The axiom of constructability, which the OP mentioned, is one such thing.

Adding the axiom of con. to ZF makes the continuum hypothesis (which is logically independent of the ZF/ZFC axioms), into a theorem. Because I don't feel any pressure to preserve anything ontologically (realism of ZFC, etc.), I don't see the harm in letting the axiom of con. in the door to turn ZFC into ZFCC. It turns the axiom of choice into a theorem and resolves many standing problems in set theory, as well. Sounds pretty good to me. But I'm not a set theoretician, or a mathematician at all, so maybe I'm not in a position to understand their reservations (and it seems like they have many). Maybe the resistance is actually to the prospect of these amendments becoming canonized in some way. But I think this actually would evidence a prevailing aversion to logical pluralism in the mathematical community, making it a cultural/ideological resistance, not a formally justified one.

But I'm mostly speculating here; does anyone want to explain to me why so many set theorists apparently resist the axiom of con. and other amendments to ZFC? Is it more ideological than anything else?

The OP is a real goldmine of talking points, but maybe we should take them one at a time to avoid monster posts so early in the thread!
I think that the primary reason the axiom of constructability is almost universally rejected is that it in some way 'misses' some of the sets. The traditional conception of the 'power set' is missing. I don't see this as a problem personally, but a lot of people take the power set operation as being a fundamental part of set theory that to remove would be unduly restrictive. Of course, there is still a notion of power set that exists (and I think it has some relation to the Henkin semantics of second order logic) but it doesn't include 'all' the subsets.

I agree that we could take these points one at a time, I guess I wasn't sure that they would all provoke interest and there is a degree of interrelatedness. Constructibility is a favourite of mine I'm more than happy to start there. I'll post something more detailed in a bit.