Practically useless math and confirmational holism

[James MC]

Dear mathematicians,

What is the most practically useless mathematical truth you know of?

Background to this question: I am evaluating an epistemological hypothesis called "confirmational holism" (http://en.wikipedia.org/wiki/Confirmation_holism). It (roughly) states that:
(i) empirical evidence only ever confirms/falsifies the conjunction of ALL of our beliefs at once
(ii) when falsification occurs we hold fixed certain conjuncts and revise others
(iii) mathematical (and logical) beliefs are typically held fixed in this process
(iv) mathematical (and logical) beliefs attain justification only by their successful role in being the conjuncts typically held fixed
(v) mathematical (and logical) truths are not justified a priori, but empirically.

I'm thinking that if there are mathematical propositions that intuitively we ahve justification for believing, that have little to no significance to physics or the empirical sciences more generally, then it could be used to problematise (iv).

Very interested in your suggestions.

 

[DonAntonio]

Dear mathematicians,

What is the most practically useless mathematical truth you know of?

Background to this question: I am evaluating an epistemological hypothesis called "confirmational holism" (http://en.wikipedia.org/wiki/Confirmation_holism). It (roughly) states that:
(i) empirical evidence only ever confirms/falsifies the conjunction of ALL of our beliefs at once
(ii) when falsification occurs we hold fixed certain conjuncts and revise others
(iii) mathematical (and logical) beliefs are typically held fixed in this process
(iv) mathematical (and logical) beliefs attain justification only by their successful role in being the conjuncts typically held fixed
(v) mathematical (and logical) truths are not justified a priori, but empirically.

I'm thinking that if there are mathematical propositions that intuitively we ahve justification for believing, that have little to no significance to physics or the empirical sciences more generally, then it could be used to problematise (iv).

Very interested in your suggestions.

Define "useless".

DonAntonio

 

[chiro]

Dear mathematicians,

What is the most practically useless mathematical truth you know of?

Background to this question: I am evaluating an epistemological hypothesis called "confirmational holism" (http://en.wikipedia.org/wiki/Confirmation_holism). It (roughly) states that:
(i) empirical evidence only ever confirms/falsifies the conjunction of ALL of our beliefs at once
(ii) when falsification occurs we hold fixed certain conjuncts and revise others
(iii) mathematical (and logical) beliefs are typically held fixed in this process
(iv) mathematical (and logical) beliefs attain justification only by their successful role in being the conjuncts typically held fixed
(v) mathematical (and logical) truths are not justified a priori, but empirically.

I'm thinking that if there are mathematical propositions that intuitively we ahve justification for believing, that have little to no significance to physics or the empirical sciences more generally, then it could be used to problematise (iv).

Very interested in your suggestions.

Hey James MC and welcome to the forums.

I see the point to your question and what you are getting at, but you should realize that in order to analyze anything at all and subsequently make sense of it (in the best way we can), then we need to introduce constraints.

Because of the above, this means that for any kind of analyses, we introduce not only constraints on our system parameters, but also constraints on our structures as well. Typically the structures are just normal numbers with a particular construction, but they can be isomorphic to many things that aren't necessarily 'visual' and have 'intuitive ordering' like points do on the real line.

Also one thing (and I think this a great thing), is that science is moving towards using uncertainty as the basis for analysis and reasoning over determinism. This is a big shift in thinking and the end result is that we make inferences and related conclusions based on the idea that have some level of 'confidence' or 'credibility' and this is very different from the idea that we are trying to find an 'exact' set of formulas to describe everything 'exactly'.

The other thing is that if you are talking about mathematics, you don't use empirical results the same way you do in the sciences. In mathematics we create the laws but in science we have to discover them and because of this, mathematicians have the luxury of proving things because they start with the egg and the scientists start with the chicken.

In terms of mathematical truths, if we find out that a particular construction leads to unlogical results, then we try a new construction. Some refer to the 'completeness' properties of such systems or axiomitized properties, but in terms of finding out absolute truth, it really depends on what you call absolute truth and what that actually refers to.

If however we have constructions that are logical, have intuitive properties (not necessary, but it helps further the idea that the construction itself yields valid properties) and have no internal contradictions, then this construction will give a level of truth related primarily to that construction. And this is often worked on empirically with the more powerful support coming from proofs that cover the largest classes of situations.

We have to fix some things and we always do (think about the constraint argument). Again if we let anything and everything happen, we wouldn't be able to make sense of it. As a result we start small and slowly expand our horizons while maintaining the constraints to be not any more general than they have to be which would result in losing the picture entirely.

 

[James MC]

DonAntonio,

Define "useless".

I'm looking for mathematical propositions debated within branches of mathematics that do not have any application in theoretical physics, and, perhaps, no conceivable application in theoretial physics (and empirical science more generally) given what we already know about the physical world.


chiro,

Thanks for the kind welcome.

but in terms of finding out absolute truth, it really depends on what you call absolute truth and what that actually refers to.

While I'm confident there is only one coherent notion of truth shared by all fields of enquiry, that is as objective as it is absolute, I would much rather stay away from this can of worms. Everything I'm interested in here can be stated without mention of truth and can be stated in terms of justified belief.

If however we have constructions that are logical, have intuitive properties (not necessary, but it helps further the idea that the construction itself yields valid properties) and have no internal contradictions, then this construction will give a level of truth related primarily to that construction. And this is often worked on empirically with the more powerful support coming from proofs that cover the largest classes of situations.

An example of such a construction would be of interest. One that is justified entirely by appeal to what you're calling "intuitive properties" and "logical coherence", and one that invokes mathematical concepts not employed by any known empirical science.

 

[SteveL27]

Dear mathematicians,

What is the most practically useless mathematical truth you know of?

Fermat's last theorem is a very well-known example. There's no practical use for Wiles's proof whatsoever, other than greatly advancing the state of knowledge of modern algebra.

You might also look into large cardinal axioms in set theory. Those are pretty useless too, if by "useful" you mean that some company's going to build a widget and make a lot of moey based on the theory.

What's your definition of utility?

 

[chiro]

While I'm confident there is only one coherent notion of truth shared by all fields of enquiry, that is as objective as it is absolute, I would much rather stay away from this can of worms. Everything I'm interested in here can be stated without mention of truth and can be stated in terms of justified belief.

But it's a subtle thing because truth is always relative to something. You need to first acknowledge the relative nature and then discuss what things are relative to.

For example, once upon a time, it was thought that everything was made out of only whole numbers and that you could not get things that were not of this form. The greatest example of this was the Pythagorean school of thought that ended up drowning one of their own members because he showed that a right-angled triangle with two unit length sides does not have a rational hypotenuse.

So the first thing you need to do is acknowledge the relativity and then in doing this you can study objective facts about some 'subset' of all that is and potentially can be.

This is why we introduce constraints in the manner I posted above.

An example of such a construction would be of interest. One that is justified entirely by appeal to what you're calling "intuitive properties" and "logical coherence", and one that invokes mathematical concepts not employed by any known empirical science.

Well we have for example constructions of sets with the ZFC theory. You have the construction of the natural numbers with the Peano-Axioms.

In terms of some of the intracacies of this, you should look at what Godels theorems (namely the incompleteness theorem) has to say about the Peano-Axioms: this will give some very specific ideas about problems faced with 'constructions' in general and I think you would benefit from reading this.

Other things include the criteria for functions to be continuous and as a further property, differentiable. This subject is one of the key foundations of mathematical analysis and it has a precise definition. You may be surprised that as far as mathematics goes, this is relatively new in terms of having rigorous definitions and this is something that is characteristic of mathematics in general.

The idea is that the properties either for the constraints (telling you if something 'is' of a particular type but not telling how specifically how to construct it) or the construction (tells you specifically how to construct the objects and system from the ground up), it needs to be mathematically 'acceptable' (usually people say rigorous) and this turns out not to be an easy thing to do.

 

[James MC]

SteveL27,

Many thanks for the interesting examples. My definition of utility is in terms of the theory I am evaluating, confirmational holism, which states that all of our beliefs are justified entirely by empirical science and empirical methods, and that there are no a priori truths, no apriori methods, and so no a priori justification for any of our beliefs.

Admittedly, confirmational holism is not well defined, and so, my definitions of utility is only going to be as clear as confirmational holism itself. Nonetheless, point (iv), and perhaps also point (i), in my initial post, is key to defining my notion of utility. For ultimately, what I'm after, are counterexamples to (iv), and perhaps (i).

Fermat's last theorem is a good case. However, a confirmational holist will say that the methods that were used to prove it were methods that have been justified by their successful use in other proofs, where the other proofs were very relevant to science.

I'm going to have to look up, and have a think about, the set theory example you gave, before responding; looks like an interesting one, thanks.

 

[James MC]

I've just found a good discussion of what I'm looking for, which is along the lines of what you both have been suggesting:

http://plato.stanford.edu/entries/mathphil-indis/#4

The relevant quote is:

Maddy's third objection is that it is hard to make sense of what working mathematicians are doing when they try to settle independent questions. These are questions, that are independent of the standard axioms of set theory — the ZFC axioms.[6] In order to settle some of these questions, new axiom candidates have been proposed to supplement ZFC, and arguments have been advanced in support of these candidates. The problem is that the arguments advanced seem to have nothing to do with applications in physical science: they are typically intra-mathematical arguments. According to [confirmational holism], however, the new axioms should be assessed on how well they cohere with our current best scientific theories. That is, set theorists should be assessing the new axiom candidates with one eye on the latest developments in physics. Given that set theorists do not do this, confirmational holism again seems to be advocating a revision of standard mathematical practice, and this too, claims Maddy, is at odds with naturalism (Maddy 1992, pp. 286–289).

So I should try to look into the arguments advanced mentioned by Maddy.

 

[SteveL27]

I've just found a good discussion of what I'm looking for, which is along the lines of what you both have been suggesting:

http://plato.stanford.edu/entries/mathphil-indis/#4

The relevant quote is:



So I should try to look into the arguments advanced mentioned by Maddy.

It's a real stretch to think that the axioms of set theory have anything to do with physics.

 

[James MC]

Hmmm, I'm not sure Steve, for example see:

http://www.sciencedirect.com/science/article/pii/S0960077996000550

I think this is why Maddy is concentrating, not on the standard axioms of set theory, but instead on these specific arguments for the introduction of new ones.

Some of this goes back to the notion of 'utility' we are working with, which is as only well defined as confirmational holism, which is not itself very well defined.

 

[SteveL27]

Hmmm, I'm not sure Steve, for example see:

http://www.sciencedirect.com/science/article/pii/S0960077996000550

I think this is why Maddy is concentrating, not on the standard axioms of set theory, but instead on these specific arguments for the introduction of new ones.

Some of this goes back to the notion of 'utility' we are working with, which is as only well defined as confirmational holism, which is not itself very well defined.

I don't have access to that article. I must say in general that it's frustrating that academic papers are behind pay firewalls. Acase can be made that taxpaying members of the public should have access to papers put out by public institutions. But that's another issue for another time.

I'd most definitely be curious to know how anyone can make a case for assigning physical meaning to the axioms of set theory. The abstract did say that most set theorists do not think there's any connection at all.

 

[James MC]

Tell me about it. If you pm your email to me I would be happy to email the article to you (incidently, at 3mb it's too large to attach to a physicsforum post).

 

[Skrew]

One of the more interesting things in axiom based mathematics is the fact that the axioms could contain an inconsistency.

This really bothered me before but I have accepted it and I consider most of the foundational mathematics to be rocksolid and unchangable.

If an inconsistency is found, all of mathematics and physics will need to be rebuilt.

It is interesting though that a lot of physics theories which deny/confirm the exsistence of things without being experimentally verified will break down.

 

[micromass]

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