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What is mathematics?


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4 hours ago, Yevgeny Karasik said:

Mathematics is the science about equal dualities. In other words, axioms of any mathematical theory can be replaced with another set of axioms, each of which asserts equivalence (or equality) of dual expressions. This fact has nothing to do with duality principles in various branches of mathematics. 

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Since you have used equality in your definition of Mathematics, what is the Science that defines equality for us, to prevent your definition becoming self referential ?

 

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  • 2 weeks later...

Physics is the study of mass, motion, and the relations between them.

Chemistry is the study of elements, compounds, and the relations between them.

Economics is the study of goods, prices and the relations between them.

...

 

Mathematics is the study of relations in the abstract.

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  • 2 months later...
On 10/25/2021 at 9:29 AM, Yevgeny Karasik said:

Mathematics is the science about equal dualities. In other words, axioms of any mathematical theory can be replaced with another set of axioms, each of which asserts equivalence (or equality) of dual expressions. This fact has nothing to do with duality principles in various branches of mathematics. 

commercial linked removed per Rule 2.7

Wow! It's a small world after all. This person used to be a classmate of mine around 50 years ago.

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Broadly speaking, mathematics is the science of how little you can assume in order to be able to say anything at all, and how much you can say after having assumed this and that.

The first one is algebra.

The second one is analysis.

<joke>

Those stand for the two A's that are in all the words mathematics, algebra, and analysis.

<end of joke>

The whole ramifications are more or less elaborate variations on these two basic themes. Some branches incorporate both.

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E. Wigner said it so well, I won't try to do better:

"I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts. ... The depth of thought which goes into the formulation of the mathematical concepts is later justified by the skill with which these concepts are used."

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40 minutes ago, Genady said:

E. Wigner said it so well, I won't try to do better:

"I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts. ... The depth of thought which goes into the formulation of the mathematical concepts is later justified by the skill with which these concepts are used."

I deeply and wholeheartedly appreciate Wigner, and he was sure much a better mathematician than I will ever be. But somehow the definition:

"mathematics is the science of skillful operations with concepts and rules invented just for this purpose (mathematics.)"

(My addition in parenthesis and my emphasis.)

leaves something wanting for me. ;) 

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9 minutes ago, joigus said:

I deeply and wholeheartedly appreciate Wigner, and he was sure much a better mathematician than I will ever be. But somehow the definition:

"mathematics is the science of skillful operations with concepts and rules invented just for this purpose (mathematics.)"

(My addition in parenthesis and my emphasis.)

leaves something wanting for me. ;) 

I understand that "this purpose" refers to "skillful operations" rather than "mathematics."

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1 minute ago, Genady said:

I understand that "this purpose" refers to "skillful operations" rather than "mathematics."

Let me try that:

I would say that mathematics is the science of skillful operations with concepts and rules invented just for the purpose of skillful operations with concepts and rules.

Nah, it doesn't work for me either. But hey, what do I know. ;) 

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7 minutes ago, joigus said:

Let me try that:

I would say that mathematics is the science of skillful operations with concepts and rules invented just for the purpose of skillful operations with concepts and rules.

 

I think this is about right:

You cannot do much with poor concepts. But invent a good concept and you can go very far by skillfully operating with it.

Actually, it is not very different from what you've said above, "how little you can assume in order to be able to say anything at all, and how much you can say after having assumed this and that." Wigner calls it "concepts" rather than "assumptions" and I agree with this.

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Yes, in fact 'assumptions' or 'concepts' go far beyond the realm of pure mathematics, so these definitions that try to be so broad really don't finish the job of specifying the matter IMO.

What I tried to do is to picture the two distinct attitudes that govern all of maths. The idea that everything in maths is either algebra or analysis is not mine AAMOF, but I can't remember where I picked it up. But the feeling, when you're doing maths, of 'I'm doing algebra' or 'I'm doing analysis' is very clear in your mind when you're doing it.

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We, i.e. Wigner and I :) , consider examples of "concepts" such as: complex number, Dirac function, Lie algebra, metric, Fourier transform, vector space, etc.

The focus in this "definition" of pure math is: a) inventing a concept, and b) skillfully operating with it to get deep and rich theorems.

I am not sure if there is a correspondence between these two parts and the two attitudes in your "definition." It doesn't have to be.

Edited by Genady
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36 minutes ago, Genady said:

vector space

A vector space is certainly a mathematical concept.

Do you know what one is ?

 

I'm sorry but I can't go along with this rule based approach you seem to have to everything in Science, including it seems, Mathematics.

It has its place, no more no less.

But there are plenty of other aspects that are not rule based.

 

1 hour ago, joigus said:

The idea that everything in maths is either algebra or analysis is not mine AAMOF, but I can't remember where I picked it up.

The Ancient Greeks thought all of Maths was founded in Geometry and this view prevailed until recently, although it was steadily eroded by other branches in the last couple of centuries.

In the early 20th century there was an attempt by certain sections to regard Algebra as the foundation, with everything being expressed in terms of Algebra.

Developements in the second half of the 20th century showed how untenable such a goal was.

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On 10/25/2021 at 9:29 AM, Yevgeny Karasik said:

Mathematics is the science about equal dualities. In other words, axioms of any mathematical theory can be replaced with another set of axioms, each of which asserts equivalence (or equality) of dual expressions. This fact has nothing to do with duality principles in various branches of mathematics. 

commercial linked removed per Rule 2.7

 

4 hours ago, Euan Taras said:

"Mathematics is the science about equal dualities", argues Yevgeny Karasik in his book "Duality Revolution" url deleted

Ha-ha-ha! @Yevgeny Karasik and @Euan Taras are one and same person! 

4 minutes ago, studiot said:

A vector space is certainly a mathematical concept.

Do you know what one is ?

 

Yes, I do.

4 minutes ago, studiot said:

I'm sorry but I can't go along with this rule based approach ...

Don't.

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8 minutes ago, Genady said:
11 minutes ago, studiot said:

A vector space is certainly a mathematical concept.

Do you know what one is ?

 

Yes, I do.

 

So why did you list the Fourier transformation as a fundamental concept ?

It is simply a skillful operation of the vector space concept.

1 hour ago, Genady said:

The focus in this "definition" of pure math is: a) inventing a concept, and b) skillfully operating with it to get deep and rich theorems.

 

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9 minutes ago, studiot said:

 

So why did you list the Fourier transformation as a fundamental concept ?

It is simply a skillful operation of the vector space concept.

 

I didn't say and didn't mean to say anything about "fundamental" concepts. Yes, some concepts are much more fundamental than others. However, concepts can also be nested.

I think that the Fourier transform concept is deeper than just a skillful operation.

There is a very good class in Stanford, EE261 - The Fourier Transform and its Applications. The first half is about the concept.

Stanford Engineering Everywhere | EE261 - The Fourier Transform and its Applications

 

Edited by Genady
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58 minutes ago, studiot said:

The Ancient Greeks thought all of Maths was founded in Geometry and this view prevailed until recently, although it was steadily eroded by other branches in the last couple of centuries.

 

Yeah, you're right. I forgot about geometry. I kind of think of geometry as a whole different class of its own. In fact geometry went through a process of trying to unify it all of its own, the Erlangen program, by Felix Klein. To me geometry is kind of a bridge between physics and pure (abstract) mathematics. But I don't know really.

I think Poincaré tried to base all of maths on the concept of group. I'm not an expert, but I don't think he was successful.

1 hour ago, Genady said:

Wigner and I :)

:) 

Poincaré and I think otherwise.  ;) And Euler agrees with us. :D

Now serious. I wish I could remember where I picked up that dichotomy into algebra and analysis. I'll look it up.

And this doesn't give me much hope:

https://math.stackexchange.com/questions/1392273/algebra-and-analysis

Edited by joigus
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13 minutes ago, joigus said:

Now serious. I wish I could remember where I picked up that dichotomy into algebra and analysis. I'll look it up.

Please, do, and let me know. I want to read more about it.

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13 hours ago, Genady said:

I didn't say and didn't mean to say anything about "fundamental" concepts. Yes, some concepts are much more fundamental than others. However, concepts can also be nested.

Well I think you implied that in your words, so I highlighted the word 'pure' - which you did use  - in my reply.

Pure Maths, by definition, does not include Applied Maths, which you also referred to in relation to 'concepts'.

14 hours ago, Genady said:

We, i.e. Wigner and I :) , consider examples of "concepts" such as: complex number, Dirac function, Lie algebra, metric, Fourier transform, vector space, etc.

The focus in this "definition" of pure math is: a) inventing a concept, and b) skillfully operating with it to get deep and rich theorems.

I am not sure if there is a correspondence between these two parts and the two attitudes in your "definition." It doesn't have to be.

I don't think I defined anything so I am not sure about your last line but taking your second line I would say that vector spaces are the "pure maths", and fourier transforms are the "skillfully operating with it to get deep and rich theorems" In other words the applied maths.

 

13 hours ago, Genady said:

There is a very good class in Stanford, EE261 - The Fourier Transform and its Applications. The first half is about the concept.

Stanford Engineering Everywhere | EE261 - The Fourier Transform and its Applications

Thank you for this reference, I expect it is an excellent course.

Edit

A quick look at this course suggests that it is pretty comprehensive.
It also states in so many words that this is an applied maths subject, firmly based in StanfordEngineering Everywhere no less.

Quote

The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used. Together with a great variety, the subject also has a great coherence, and the hope is students come to appreciate both

Lecture 1 refers to 'analysis' and 'synthesis', a subject distinction you to seem to wish to avoid.

 

When I studied the Integral transforms in general I was following a part of my math degree called 'Linear Analysis' which had both pure and applied components.

The pure componet was represented in the course texts by Nering and by Hoffman and Kunze.
The applied component was represented by the text by Keider, Kuller, Ostberg and Perkins.

Needless to say the integral transforms came in the applied component.

 

However my main objection to your 'rule based' approach is that it is 'static'.
Adopting this therefore precludes the study of process in Mathematics, process being a dynamic object.

This approach is therefore akin to studying only statics in Mechanics, and ignoring dynamics and all that dynamics introduces.

 

Finally another member recently tried to categories the parts of Mathematics, you may wish to look at their thread.

 

Edited by studiot
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5 minutes ago, studiot said:

I don't think I defined anything so I am not sure about your last line but taking your second line I would say that vector spaces are the "pure maths", and fourier transforms are the "skillfully operating with it to get deep and rich theorems" In other words the applied maths.

  • "Your 'definition'" in my post above was in reply to the previous post by @joigus, not @studiot.
  • I don't think that pure mathematical theorems such as Pythagoras is applied math.
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29 minutes ago, Genady said:

I don't think that pure mathematical theorems such as Pythagoras is applied math.

I don't think anyone suggested they are applied maths.

32 minutes ago, Genady said:

Your 'definition'" in my post above was in reply to the previous post by @joigus, not @studiot.

Fair enough, not me then.

 

Now how about properly addressing my points and making a discussion of it ?

 

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50 minutes ago, studiot said:

Well I think you implied that in your words, so I highlighted the word 'pure' - which you did use  - in my reply.

Pure Maths, by definition, does not include Applied Maths, which you also referred to in relation to 'concepts'.

I don't think I defined anything so I am not sure about your last line but taking your second line I would say that vector spaces are the "pure maths", and fourier transforms are the "skillfully operating with it to get deep and rich theorems" In other words the applied maths.

 

Thank you for this reference, I expect it is an excellent course.

Edit

A quick look at this course suggests that it is pretty comprehensive.
It also states in so many words that this is an applied maths subject, firmly based in StanfordEngineering Everywhere no less.

Lecture 1 refers to 'analysis' and 'synthesis', a subject distinction you to seem to wish to avoid.

 

When I studied the Integral transforms in general I was following a part of my math degree called 'Linear Analysis' which had both pure and applied components.

The pure componet was represented in the course texts by Nering and by Hoffman and Kunze.
The applied component was represented by the text by Keider, Kuller, Ostberg and Perkins.

Needless to say the integral transforms came in the applied component.

 

However my main objection to your 'rule based' approach is that it is 'static'.
Adopting this therefore precludes the study of process in Mathematics, process being a dynamic object.

This approach is therefore akin to studying only statics in Mechanics, and ignoring dynamics and all that dynamics introduces.

 

Finally another member recently tried to categories the parts of Mathematics, you may wish to look at their thread.

 

"Now how about properly addressing my points and making a discussion of it ?"

Let's do.

  • "Well I think you implied that in your words..." -- This is in reference to "fundamental". No, I did not.
  • "Pure Maths, by definition, does not include Applied Maths, which you also referred to in relation to 'concepts'." -- Agree with the first half. Regarding the second part, No, I did not.
  • "... "skillfully operating with it to get deep and rich theorems" In other words the applied maths." -- This, as you said in your last post, "suggested they are applied maths." They certainly are not.
  • The whole section about the Fourier transform. -- You don't like to call it a concept. Just cross it off my list of examples. It is not too dear to me.
  • "Lecture 1 refers to 'analysis' and 'synthesis'..." -- It refers specifically to the direct and inverse Fourier transforms.
  • "... my main objection to your 'rule based' approach ..." -- I don't find "my approach" to be an interesting subject for discussion.

 

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12 minutes ago, Genady said:

... "skillfully operating with it to get deep and rich theorems" In other words the applied maths." -- This, as you said in your last post, "suggested they are applied maths." They certainly are not.

So you disagree with the quoted statement from your own link ?

13 minutes ago, Genady said:

Lecture 1 refers to 'analysis' and 'synthesis'..." -- It refers specifically to the direct and inverse Fourier transforms.

It refers to manynthings.

Please don't sidestep the issue by implying that there it refers to only one matter.

Did you ever take this course ?

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1 minute ago, studiot said:

So you disagree with the quoted statement from your own link ?

No, I disagree with your addition to my statement, namely "In other words the applied maths."

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