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What is mathematics?


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Just now, studiot said:

Then you were not clear enough.

And I apologize for that.

4 minutes ago, studiot said:

It refers to many things.

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

Did you ever take this course ?

Yes, I took the course. I know that it was the only thing the prof referred to in Lecture 1.

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2 minutes ago, Genady said:

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

I did indeed state that fourier transforms are applied maths.

You have contradicted that statement more than once now.

I have also pointed out that the very words in the link you provided state explicitly that fourier trnasforms are applied maths.

I even quote the passage from that link.

Yet you seem to maintain that fourier transforms are not applied maths.

 

Have you ever studied fourier transforms at all ?

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3 minutes ago, studiot said:

I did indeed state that fourier transforms are applied maths.

You have contradicted that statement more than once now.

I have also pointed out that the very words in the link you provided state explicitly that fourier trnasforms are applied maths.

I even quote the passage from that link.

Yet you seem to maintain that fourier transforms are not applied maths.

 

Have you ever studied fourier transforms at all ?

I have already answered the point regarding the Fourier transform. Copying it here:

  • 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.

I will not reply any more to your questions about me, my knowledge, my education, etc. It is not your business, neither it is the purpose of the forums.

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Studiot

1 hour ago, studiot said:

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

Genady

28 minutes ago, Genady said:

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

This most definitely appears to contradict my comment

yet

Stanford

Quote

Topics: Previous Knowledge Recommended (Matlab), The Fourier Series, Analysis V. Synthesis, Periodic Phenomena And The Fourier Series -Periodicity In Time And Space -Reciprocal Relationship Between Domains, The Reciprocal Relationship Between Frequency And Wavelength

 

I have underlined the relevant words in the topics list appearing under lecture 1.

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1 minute ago, studiot said:

Studiot

Genady

This most definitely appears to contradict my comment

yet

Stanford

 

I have underlined the relevant words in the topics list appearing under lecture 1.

Now go ahead and listen what he says about this topic.

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5 minutes ago, Genady said:

Now go ahead and listen what he says about this topic.

Yes, and when I do (it's nearly one hour long) I will not misrepresent what he (or anyone else) says, and I would expect to learn something I did not already know, even though it was only lecture 1 of 30.

I have already said that it seems a good course, and its also free.

I note from the transcript that both analysis and synthesis are defined and contrasted.

Edited by studiot
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16 minutes ago, studiot said:

Yes, and when I do (it's nearly one hour long) I will not misrepresent what he (or anyone else) says, and I would expect to learn something I did not already know, even though it was only lecture 1 of 30.

I have already said that it seems a good course, and its also free.

I note from the transcript that both analysis and synthesis are defined and contrasted.

The entire Analysis vs. Synthesis topic in this lecture takes 2 minutes, between 21:00 and 23:00. You can jump straight there:

Stanford Engineering Everywhere | EE261 - The Fourier Transform and its Applications | Lecture 1 - Previous Knowledge Recommended (Matlab)  

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5 minutes ago, Genady said:

The entire Analysis vs. Synthesis topic in this lecture takes 2 minutes, between 21:00 and 23:00. You can jump straight there:

Stanford Engineering Everywhere | EE261 - The Fourier Transform and its Applications | Lecture 1 - Previous Knowledge Recommended (Matlab)  

 

So just as I said Lecture 1 contains many things, including analysis and synthesis.

 

Edited by studiot
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From a newbie point of view .

Mathematics is simply the arrangement of numbers and unknowns

i can start to learn a lot of beginner mathematics if i keep that point in my mind

 

😅

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On 1/9/2022 at 11:35 PM, Genady said:

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

I don't seem to be able to find it now. It must have been a university professor.

What I have been able to check though is that many people seem to factually use that dichotomy of mathematics into analysis and algebra, because there are many questions concerning it.

It's true that geometry is a completely different animal.

I would say that the starting point for the three of them is axioms --that's just what mathematics does. But both analysis and algebra posit them as identities based on definitions, while geometry posits them as giving rise to formulas that claim to state relations in a world of 'visual entities' (Pythagorean theorem, Thales' theorem etc.) Those belong in the realm of perception, or intuitions about space, I would say.

From these reasonable intuitions, you are compelled to transform them into algebraic statements that you later use to derive further theorems, and solve problems. 

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1 hour ago, joigus said:

geometry ... belong in the realm of perception, or intuitions about space, I would say.

It looks like concepts in geometry are idealizations of perceptions and intuitions about space, concepts in algebra are idealizations of perceptions and intuitions about counting, and concepts in analysis are idealizations of perceptions and intuitions about acting.

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1 hour ago, Genady said:

It looks like concepts in geometry are idealizations of perceptions and intuitions about space, concepts in algebra are idealizations of perceptions and intuitions about counting, and concepts in analysis are idealizations of perceptions and intuitions about acting.

What is the value of zero? 

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24 minutes ago, dimreepr said:

What is the value of zero? 

In geometry, there is no zero. That's either algebra or analysis.

Edited by joigus
minor addition
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1 hour ago, joigus said:

In geometry, there is no zero. That's either algebra or analysis.

Euclid

Definition#1   "That which hath no part."

 

This is an early version of a metric which you cannot have without a zero.

Edited by studiot
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As a physicist, I'm not overly concerned with the 'why' of a paradigm, or set of rules, only its 'function'.

Math is simply a tool for describing/investigating the world around us.
No differentthan a hammer, or a pen; but much, much more useful.

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5 hours ago, dimreepr said:

What is the value of zero? 

zero/0 obviously has a value...a zero value, but a value none the less. eg: My bank account may have a value of zero dollars. But give me a cheque worth $1,000,000, and you can shove in as many zeros as you like. In essence, mathematics, the language of physics, does not exist without zero value.

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5 hours ago, studiot said:

Euclid

Definition#1   "That which hath no part."

 

Ok. I trust you and Euclid. But keep in mind that when Euclid wrote his Elements, there was no distinction analysis/algebra/geometry, so when he wrote that, he didn't make a clear distinction was probably trying to introduce the minimal elements of analysis necessary for doing geometry. Obviously you cannot do anything at all in maths if you don't start out with some elements of algebra and analysis. But zero is the distance between two points only when they're the same point, and as for a coordinate, it doesn't mean anything that its value happens to be zero. So I don't think zero is a relevant part of geometry. There is no 'zero point,' as opposed to the real number zero in analysis, or the element zero in a ring (algebra), etc. That was kind of my point --no pun intended.

Edit (addition): On the other hand..., 'that which hath no part.' I don't know what to do with that. I don't think Euclid was in his finest hour when he wrote that.

Edited by joigus
significant qualification
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2 hours ago, joigus said:

Ok. I trust you and Euclid. But keep in mind that when Euclid wrote his Elements, there was no distinction analysis/algebra/geometry, so when he wrote that, he didn't make a clear distinction was probably trying to introduce the minimal elements of analysis necessary for doing geometry. Obviously you cannot do anything at all in maths if you don't start out with some elements of algebra and analysis. But zero is the distance between two points only when they're the same point, and as for a coordinate, it doesn't mean anything that its value happens to be zero. So I don't think zero is a relevant part of geometry. There is no 'zero point,' as opposed to the real number zero in analysis, or the element zero in a ring (algebra), etc. That was kind of my point --no pun intended.

Edit (addition): On the other hand..., 'that which hath no part.' I don't know what to do with that. I don't think Euclid was in his finest hour when he wrote that.

 

2 hours ago, joigus said:

But zero is the distance between two points only when they're the same point,

But that is part of the definition of a distance function (the Mathematical term for a metric)

Without zero you have no metric.

But that is not the only use of zero in Mathematics.

You probably know the four colour theorem, and the two colour theorem.

Can you draw a map with zero colours ?

Or perhaps you would like this poem

Quote

Sir Frederick Soddy  1936.

For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.
Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum

I have highlighted the use of zero bend (curvature or the reciprocal of radius) to indicate a straight line.

You also need zero in projective geometry for the ratio theorem to indicate the 'missing' ratio.

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9 minutes ago, studiot said:

 

But that is part of the definition of a distance function (the Mathematical term for a metric)

Without zero you have no metric.

But that is not the only use of zero in Mathematics.

You probably know the four colour theorem, and the two colour theorem.

Can you draw a map with zero colours ?

Or perhaps you would like this poem

I have highlighted the use of zero bend (curvature or the reciprocal of radius) to indicate a straight line.

You also need zero in projective geometry for the ratio theorem to indicate the 'missing' ratio.

Loved the poem. Thank you.

Somehow I don't see zero so much as a geometric concept. I think it's more of an algebraic concept. You can do quite a bit of geometry without it (similarity of triangles, Thales theorem, Pythagorean theorem, and so on.)

In a curvilinear space you don't really have zero curvature; and in a pseudo Euclidean space there are infinitely many points that have zero "distance" with respect to any one point. I think it's more of an auxiliary concept than really central to geometry. You can do some geometry without mentioning zero. You can't really start doing algebra or analysis without it.

7 hours ago, MigL said:

As a physicist, I'm not overly concerned with the 'why' of a paradigm, or set of rules, only its 'function'.

Math is simply a tool for describing/investigating the world around us.
No differentthan a hammer, or a pen; but much, much more useful.

I don't disagree with this at all. Maths is a tool. And we'd rather use maths to make a hammer than use hammers to do maths. I even think maths is at the basis of language. Even people who say they hate maths, I think, have a simpler, more basic way of mathematically understanding the world. Perhaps less sophisticated, refined, or whatever.

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17 minutes ago, joigus said:

I even think maths is at the basis of language. Even people who say they hate maths, I think, have a simpler, more basic way of mathematically understanding the world. Perhaps less sophisticated, refined, or whatever.

If I remember correctly, we have an area in the brain which is activated for basic counting - small quantities, comparing sizes, amounts, etc. This area is separate and removed from the language areas, and the two get activated independently in different situations / tasks. This does not support the view that math is at the basis of language.

In addition to this, the quantifying systems and languages come in a variety of sophistications in different cultures. In some, language is very rich and complex, while counting is very limited. In others, vice versa. Again, not a refutation, but doesn't support the hypothesis.

Edited by Genady
wrong word
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10 hours ago, joigus said:

Somehow I don't see zero so much as a geometric concept. I think it's more of an algebraic concept. You can do quite a bit of geometry without it (similarity of triangles, Thales theorem, Pythagorean theorem, and so on.)
In a curvilinear space....

 

And yet, as I noted, it is the first statement to appear in Euclid!

You can't have Pythagoras, without algebra.

I don't see how much Geometry you can do without numbers.

There is no similarity theorem in curvilinear space eg in spherical triangles.

You can't have vector geometry without the zero vector as it is involved in four of the vector axioms.

Therefore the link between extrinsic and intrinsic geometry (holonomy) fails as you can't have vector fields without vectors.

Without a side of zero length, Aubel's theorem in plane geometry is incomplete.

 

What I am saying is that 'zero' is buried very deep in Geometry so that we don't normally think about it,

 

On the other hand I fully support this comment.

10 hours ago, joigus said:

I even think maths is at the basis of language.

There is plenty of archeological evidence that maths was in at the beginning of language. (see John Derbyshire "Unknown Quantity").
However perhaps 'counting' is too sophisticated a word and a deeper, but incredibly useful, mathematical technique of tallying (putting in one-to-one correspondence) came first.
Perhaps this process even came before real words separated out from grunts which simply meant "Attention!"

 

10 hours ago, joigus said:
18 hours ago, MigL said:

As a physicist, I'm not overly concerned with the 'why' of a paradigm, or set of rules, only its 'function'.

Math is simply a tool for describing/investigating the world around us.
No differentthan a hammer, or a pen; but much, much more useful.

I don't disagree with this at all. Maths is a tool. And we'd rather use maths to make a hammer than use hammers to do maths. I even think maths is at the basis of language. Even people who say they hate maths, I think, have a simpler, more basic way of mathematically understanding the world. Perhaps less sophisticated, refined, or whatever.

May I respectfully remind both you and @MigL that this thread was started by a professor of Mathematics, in the Mathematics section ?
Whilst I would wholeheartedly agree that maths is an incredibly useful tool in many disciplines, especially Physics, this is surely a question of Mathematics?

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51 minutes ago, studiot said:

You can't have Pythagoras, without algebra.

I don't see how much Geometry you can do without numbers.

But I don't disagree with this. In fact:

15 hours ago, joigus said:

Obviously you cannot do anything at all in maths if you don't start out with some elements of algebra and analysis.

Let me add another definition of maths that I've heard to Marcus du Sautoy, if I remember correctly:

Maths is the study of patterns

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