What is "good" mathematics?

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heracleitos
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What is "good" mathematics?

Post by heracleitos »

Good mathematics is meaningless

https://en.m.wikipedia.org/wiki/Abstraction_(mathematics)
Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected.
Mathematics must have no connection whatsoever with the real world. Therefore, good mathematics is fundamentally meaningless.

Good mathematics is useless

Godfrey Harold Hardy, English mathematician wrote:
I have never done anything "useful". No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.

We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.
Claims that are meaningless are in principle also useless. It is a rather unwanted accident if these claims somehow turn out to be useful anyway. By design, they should not.

Good mathematics is ridiculous

There must still be some non-trivial gap between premises and conclusion.

In Critique of Pure Reason, Immanuel Kant distinguishes:

- analytic proposition: a proposition whose predicate concept is contained in its subject concept.

- synthetic proposition: a proposition whose predicate concept is not contained in its subject concept but related.

A mathematical theorem must be synthetic. It must argue a proposition that requires non-trivial symbol manipulation in order to derive it from its premises.

Therefore, legitimate mathematics must produce a meaningless and useless statement that is still different from the meaningless and useless statements from which it has been derived.

Hence, a good mathematical statement is ridiculous.

This is probably the reason why category theory is widely considered to be the flagship of mathematics:
Wikipedia on "abstract nonsense" wrote: In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory and homological algebra.

Labeling an argument "abstract nonsense" is usually not intended to be derogatory,[1][2] and is instead used jokingly,[3] in a self-deprecating way,[4] affectionately,[5] or even as a compliment to the generality of the argument.

When an audience can be assumed to be familiar with the general form of such arguments, mathematicians will use the expression "Such and such is true by abstract nonsense" rather than provide an elaborate explanation of particulars.[2]

The term predates the foundation of category theory as a subject itself. Referring to a joint paper with Samuel Eilenberg that introduced the notion of a "category" in 1942, Saunders Mac Lane wrote the subject was 'then called "general abstract nonsense"'.
Conclusion

The true nature of mathematics is that it is meaningless and useless. Its only redeeming quality is that it is also ridiculous.

That is in fact the very reason why I like mathematics. It satisfies my need to indulge in absurdity.
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Sculptor1
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Re: What is "good" mathematics?

Post by Sculptor1 »

heracleitos wrote: April 30th, 2022, 9:53 pm Good mathematics is meaningless

https://en.m.wikipedia.org/wiki/Abstraction_(mathematics)
Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected.
Mathematics must have no connection whatsoever with the real world. Therefore, good mathematics is fundamentally meaningless.

Good mathematics is useless

Godfrey Harold Hardy, English mathematician wrote:
I have never done anything "useful". No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.

We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.
Claims that are meaningless are in principle also useless. It is a rather unwanted accident if these claims somehow turn out to be useful anyway. By design, they should not.

Good mathematics is ridiculous

There must still be some non-trivial gap between premises and conclusion.

In Critique of Pure Reason, Immanuel Kant distinguishes:

- analytic proposition: a proposition whose predicate concept is contained in its subject concept.

- synthetic proposition: a proposition whose predicate concept is not contained in its subject concept but related.

A mathematical theorem must be synthetic. It must argue a proposition that requires non-trivial symbol manipulation in order to derive it from its premises.

Therefore, legitimate mathematics must produce a meaningless and useless statement that is still different from the meaningless and useless statements from which it has been derived.

Hence, a good mathematical statement is ridiculous.

This is probably the reason why category theory is widely considered to be the flagship of mathematics:
Wikipedia on "abstract nonsense" wrote: In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory and homological algebra.

Labeling an argument "abstract nonsense" is usually not intended to be derogatory,[1][2] and is instead used jokingly,[3] in a self-deprecating way,[4] affectionately,[5] or even as a compliment to the generality of the argument.

When an audience can be assumed to be familiar with the general form of such arguments, mathematicians will use the expression "Such and such is true by abstract nonsense" rather than provide an elaborate explanation of particulars.[2]

The term predates the foundation of category theory as a subject itself. Referring to a joint paper with Samuel Eilenberg that introduced the notion of a "category" in 1942, Saunders Mac Lane wrote the subject was 'then called "general abstract nonsense"'.
Conclusion

The true nature of mathematics is that it is meaningless and useless. Its only redeeming quality is that it is also ridiculous.

That is in fact the very reason why I like mathematics. It satisfies my need to indulge in absurdity.
Whatever it might be, maths is useful not useless. And whilst that is true it matters very little what you think is the "true nature"
Magnus Anderson
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Re: What is "good" mathematics?

Post by Magnus Anderson »

That's quite a lot to digest. You made a number of claims, five in fact, and I don't know where to start.

1) Mathematics must have no connection whatsoever with the real world.

2) Good mathematics is meaningless.

3) Good mathematics is useless.

4) Good mathematics is ridiculous.

5) Most, if not all, mathematical statements are synthetic statements.

Perhaps you should pick one -- the one that you think is the most important one -- and go from there. It's a lot easier to manage the discussion that way.

Right now, I will do no more than address your number five. You define the word "synthetic" to mean "a proposition whose predicate concept is not contained in its subject concept but related". Can you explain what it means for one concept to be contained within another? For example, what does it mean for the concept "unmarried" to be contained within the concept "bachelor"? My own understanding, which might be wrong given that I never read Kant, is that it means that the meaning assigned to the word "bachelor" is such that everything that can be represented by the word "bachelor" can also be represented by the word "unmarried". An analytic statement would thus be one that is true or false by definition. "All bachelors are married" would be an example. But here's the thing: the same applies to "5 + 7 = 12". The predicate ("12") is fully contained within the subject ("5 + 7".) Yet, according to Kant, this statement is a synthetic one; and the often stated reason is that the concept "12" isn't contained within "5", "+" and "7".
heracleitos
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Re: What is "good" mathematics?

Post by heracleitos »

Magnus Anderson wrote: May 1st, 2022, 12:23 pm An analytic statement would thus be one that is true or false by definition. "All bachelors are married" would be an example. But here's the thing: the same applies to "5 + 7 = 12". The predicate ("12") is fully contained within the subject ("5 + 7".) Yet, according to Kant, this statement is a synthetic one; and the often stated reason is that the concept "12" isn't contained within "5", "+" and "7".
As I understand Kant's analytic-synthetic distinction, in the context of mathematics, a synthetic statement has a justification, while an analytic statement is a foundationalist first principle. So, a theorem is synthetic while an axiom is analytic.

In that view, arithmetic theory (PA) is the analytic context, while the statement "5 + 7 = 12" as a theorem has a justification distinct from itself, i.e. a proof, that connects this statement to the theory.

Apparently, Kant (more or less) concurred but Frege did not see it that way.
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UniversalAlien
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Re: What is "good" mathematics?

Post by UniversalAlien »

Yes, it all adds up.

But remember 0 +{-}{\} 0 = 0

But add a one [1] and you have the World :roll:
Magnus Anderson
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Re: What is "good" mathematics?

Post by Magnus Anderson »

heracleitos wrote: May 1st, 2022, 1:11 pmAs I understand Kant's analytic-synthetic distinction, in the context of mathematics, a synthetic statement has a justification, while an analytic statement is a foundationalist first principle. So, a theorem is synthetic while an axiom is analytic.
Wouldn't that also mean that "All bachelors are married" is a synthetic statement? Do you believe that people are justified in holding that belief if they used no reasoning -- no deductive logic, for example -- to arrive at it?
heracleitos
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Re: What is "good" mathematics?

Post by heracleitos »

Magnus Anderson wrote: May 2nd, 2022, 12:24 pm Wouldn't that also mean that "All bachelors are married" is a synthetic statement? Do you believe that people are justified in holding that belief if they used no reasoning -- no deductive logic, for example -- to arrive at it?
For natural language, in my opinion, a statement is analytic, if all you need in order to verify its truth, is a dictionary. This may indeed require some incidental use of logic. I believe that this is what Kant meant to say.

Natural language is not a formal system. Verifying the truth of "All bachelors are married" can be achieved by using just its dictionary entry. I think that some use of logic is allowed for verifying analytic natural language statements in order to bridge issues caused by their informal nature.

For mathematics, on the other hand, the distinction between synthetic and analytic will always have to be somehow compatible with its fundamental epistemology, i.e. proof theory. Therefore, the slightest use of logic would already turn a first principle into a theorem.

If you use proof assistant software, however, to verify the truth of "All bachelors are married", then it becomes a mathematical situation, in which the software will have to produce or at least formally verify the proof. The necessity of a proof and verification steps would turn the statement into a synthetic one.

According to the epistemology of mathematics, every logic sentence that has an associated proof is a synthetic one.
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Sy Borg
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Re: What is "good" mathematics?

Post by Sy Borg »

What is the difference between "good mathematics" and "bad mathematics"?

The OP argument seems more coherent if talking about "pure mathematics". While most pure mathematics models are more like works or art than models of reality, there have been times when pure mathematics have predicted physical phenomena, eg. antimatter.
heracleitos
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Re: What is "good" mathematics?

Post by heracleitos »

Sy Borg wrote: May 2nd, 2022, 8:59 pm What is the difference between "good mathematics" and "bad mathematics"?
Godrey Hardy distinguished between "trivial" and "real" mathematics:
Hardy on "real" mathematics wrote: We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.
It is actually the same distinction as between "good" and "bad" mathematics.

In "trivial" mathematics there is still some connection with the physical universe. The abstraction process is considered to be incomplete:
Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected.
In "real" or "good" mathematics, the connection has been successfully and completely severed.
Sy Borg wrote: May 2nd, 2022, 8:59 pm The OP argument seems more coherent if talking about "pure mathematics".
As Hardy wrote, "real" (or "good") mathematics is "pure", i.e. pure of any possible connection to the physical universe. So, yes, purity is an important goal in mathematics. Impure mathematics is not "real" mathematics.

In this view, "applied" mathematics, such as in science and engineering, is not "real" mathematics. Instead, it is science or engineering.

These disciplines obviously have their own merit. However, because of their correspondentist notions of truth, they do something completely different from mathematics.

They are not mathematicians, but users of mathematics, who use it as a tool to maintain consistency in what they proclaim about the physical universe.
Sy Borg wrote: May 2nd, 2022, 8:59 pm While most pure mathematics models are more like works or art than models of reality, there have been times when pure mathematics have predicted physical phenomena, eg. antimatter.
Scientists and engineers do "useful work" but that is not a compliment in mathematics. Since they seek to connect to the physical universe, describe it, alter it, and since even consider the physical universe to be their benchmark for truth, they are on a completely different road. The usefulness of their work certainly has merit, but not in the domain of "real" or "good" mathematics.

If scientists introduce pure mathematics into their work on the physical universe, then they have managed to successfully use it as a tool. There is nothing wrong with that. Mathematics itself, however, would never do that, because in pure mathematics the physical universe gets resolutely rejected as a source of truth.
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Sy Borg
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Re: What is "good" mathematics?

Post by Sy Borg »

Thanks. I didn't know of mathematics that determinedly avoids the physical, only that which doesn't concern itself either way.

I don't have a problem with it, just as I have no problem with esoteric scientific research. It disappoints me that blue skies research is out of fashion, as compared with strictly practical and commercial projects. Sometimes the most extraordinary discoveries when a scientist is pursuing a line of inquiry out of sheer fascination, following the clues wherever they may lead. Many discoveries in history have stemmed from unguided exploration.

Still, those paying the grants and salaries require a return on their investments, and blue skies research and pure math would be seem as a gambler's option, incurring opportunity costs. Esoteric research and pure math have a relatively high chance of finding nothing that is useful and a small chance of making huge returns.
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