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Cuchulainn
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Re: Philosophy of Mathematics

January 15th, 2018, 8:46 am

``Can you write your question in maths or give a concrete case? It's a bit fuzzy."

you are using fuzzy logic, my question is crystal clear! Try Boolean !

anyway thanks for all the input! :+)
OK, you have answered your own question.

On 'negative' probabilities, we have not seen the great concrete example, as Paul Halmos would say.
After all, it is the philosophy of mathematics thread..
 
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Collector
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Re: Philosophy of Mathematics

January 15th, 2018, 4:14 pm

``Can you write your question in maths or give a concrete case? It's a bit fuzzy."

you are using fuzzy logic, my question is crystal clear! Try Boolean !

anyway thanks for all the input! :+)
OK, you have answered your own question.

On 'negative' probabilities, we have not seen the great concrete example, as Paul Halmos would say.
After all, it is the philosophy of mathematics thread..
yes binary is possibly the way to go in this context.

"Boolean algebra was introduced in... 1847". (wiki) this must be joke!
 
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Cuchulainn
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Re: Philosophy of Mathematics

January 16th, 2018, 2:02 pm

I think I missed the punch-line(s).
 
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Collector
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Re: Philosophy of Mathematics

January 16th, 2018, 7:35 pm

I think I missed the punch-line(s).
you just missed the Nothing(s) between the punch-line(s)! 

"Nothing dose not exist. It allows existence." L.

What is Math without Nothing = then it has the silly zero! Just ask Pythagoras! Without Nothing there would be no math and no silly zero. Luckily without zero we still have nothing and nothing have us!
 
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DrBen
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Re: Philosophy of Mathematics

February 5th, 2018, 8:21 pm

Putting conventions aside (i.e. base 10 versus another number, 24 hours a day and so on), and underlying philosophical explanation for why mathematics itself (i.e. ideas not conventions if that makes sense) reflects the reality we find ourselves has a long and rich history. Plato's Theory of Forms, with its land of pure forms and when we 'do' mathematics we perceive a shadow cast into our world, In fact, I take the same viewpoint with the study of risk which I see as some higher dimensional form, where us as parties in quantitative finance perceive merely shadows of this ideal. No one model or method can encapsulate this form yet each reflects a shadow of this form cast into the reality we exist in. A risk manager capture one aspect, a stat arb trader another and so on.... That is, maths fits the reality we find ourselves in because our understanding of and the world itself are shadows of these ideal forms.

This aside the work  of Réné Descartes I would also cite, in particular his Method and Meditations. In particular, the Methods describe the scientific method which is in words of Plato, would be the means by which we can perceive the shadows cast by the ideal forms.....
Last edited by DrBen on February 5th, 2018, 9:16 pm, edited 2 times in total.
 
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DrBen
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Re: Philosophy of Mathematics

February 5th, 2018, 8:22 pm

Duplicate...
Last edited by DrBen on February 5th, 2018, 8:57 pm, edited 3 times in total.
 
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Collector
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Re: Philosophy of Mathematics

February 6th, 2018, 9:13 am

Putting conventions aside (i.e. base 10 versus another number, 24 hours a day and so on), and underlying philosophical explanation for why mathematics itself (i.e. ideas not conventions if that makes sense) reflects the reality we find ourselves has a long and rich history. Plato's Theory of Forms, with its land of pure forms and when we 'do' mathematics we perceive a shadow cast into our world, In fact, I take the same viewpoint with the study of risk which I see as some higher dimensional form, where us as parties in quantitative finance perceive merely shadows of this ideal. No one model or method can encapsulate this form yet each reflects a shadow of this form cast into the reality we exist in. A risk manager capture one aspect, a stat arb trader another and so on.... That is, maths fits the reality we find ourselves in because our understanding of and the world itself are shadows of these ideal forms.

This aside the work  of Réné Descartes I would also cite, in particular his Method and Meditations. In particular, the Methods describe the scientific method which is in words of Plato, would be the means by which we can perceive the shadows cast by the ideal forms.....
Plato's theory of Forms, is this not the very moment when things started to go bad and we ended up in the epicycles of the modern world, where endless fudges are used to cover errors in the very fundament of modern platonic thinking ?

What kind of geometry did Plato use for his theory of Forms, well lets ask Schrödinger:

``Democritus was intensely interested in geometry, not as a mere enthusiast like Plato; he was a geometer of distinction.  — Schrödinger 1954 

and Plato supposedly wanted to burn as many of Democritus books as possible. The glorious Pythagoreans Amyclas and Cleinas supposedly tried to persuaded Plato that the burning was pointless as Democritus' books were already in wide circulation.  But it looks like Plato and his followers succeeded, because why are there not one single copy left of Democritus brilliant work (a little paradox here?). (I think possibly one copy of Democritus brilliant work survived, but where is it? well guarded I am sure)

Since I mentioned the glorious Pythagoreans. Pythagoras theorem holds under Democritus geometric thinking (down to a sub-subatomic level where it breaks down, but not causing problems before that), but breaks down for example under ideas of discrete space, as very well pointed out by  Hermann Weyl. The Weyl tile argument.

And many mathematicians mislike bringing any physics into math? (partly understandable) But also possibly  another reason why possibly parts of the math fundament is non optimal ?  Physics and math should ideally work hand in hand? Discoveries in math feeding back in physics and discoveries in physics feeding back in math.   Or is this my misconception?

?! (factorial question mark = more questions than answers )
 
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Cuchulainn
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Re: Philosophy of Mathematics

February 6th, 2018, 12:08 pm

Discoveries in math feeding back in physics and discoveries in physics feeding back in math.   
Sometimes.
A good example the delta function which works in practice (handwaving) but was not proved in theory. We had to wait for Laurent Schwarz and S.L. Sobolev to lay the mathematical foundations which to yuge number of applications.

e.g PDE

http://math.uchicago.edu/~may/REU2014/R ... llerin.pdf


And physicists commandeer mathematicians' work all the time. You can't blame them I suppose. At other times they might take liberties and stretch definitions almost to breaking point.

// Don't forget Archimedes, no better mathematician than he, ever.
 
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Collector
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Re: Philosophy of Mathematics

February 6th, 2018, 1:04 pm

Discoveries in math feeding back in physics and discoveries in physics feeding back in math.   
Sometimes.
A good example the delta function which works in practice (handwaving) but was not proved in theory. We had to wait for Laurent Schwarz and S.L. Sobolev to lay the mathematical foundations which to yuge number of applications.

e.g PDE

http://math.uchicago.edu/~may/REU2014/R ... llerin.pdf


And physicists commandeer mathematicians' work all the time. You can't blame them I suppose. At other times they might take liberties and stretch definitions almost to breaking point.

// Don't forget Archimedes, no better mathematician than he, ever.
Arcemedies work treated almost as bad as the atomists

Archimedes Palimpsest ``It was overwritten with a Christian religious text by 13th-century monks.``

Things went pretty bad from 13th-centurty, and I am not even sure the Dark age is over. I just got hold of some original documents showing certain schools of wisdom were treated pretty bad at least until late 1800.
 
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Cuchulainn
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Re: Philosophy of Mathematics

May 23rd, 2018, 7:02 pm

Cauchy invented the delta function 100 years before Dirac.


[$]\delta(t) = \displaystyle\lim_{\varepsilon\to \:  0} \varepsilon^2/(\varepsilon^2 + t^2)[$]
Last edited by Cuchulainn on May 23rd, 2018, 7:22 pm, edited 5 times in total.
 
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Traden4Alpha
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Re: Philosophy of Mathematics

May 23rd, 2018, 7:12 pm

Cauchy invented the delta function 100 years before Dirac.



[$]\delta(t) = \displaystyle\lim_{\varepsilonto\infty} (\varepsilon)^{2}/(\varepsilon^2 + t^2}[$]
Correcting this injustice is a step in the right direction.
 
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Cuchulainn
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Re: Philosophy of Mathematics

May 23rd, 2018, 7:18 pm

All was not lost; at the time Schwartz, Sobolev and others took the delta function by the scruff of the neck and generalize to the theory of Distributions, without which there would be no Finite Element Method.

The concrete example again..

So, instead of huffing and puffing with Fourier and integral, just go for the Cauchy distribution?
 
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Traden4Alpha
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Re: Philosophy of Mathematics

May 23rd, 2018, 8:02 pm

All was not lost; at the time Schwartz, Sobolev and others took the delta function by the scruff of the neck and generalize to the theory of Distributions, without which there would be no Finite Element Method.

The concrete example again..

So, instead of huffing and puffing with Fourier and integral, just go for the Cauchy distribution?
Throwing Cauchy'n to the wind, what?
 
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Collector
Posts: 3965
Joined: August 21st, 2001, 12:37 pm

Re: Philosophy of Mathematics

May 25th, 2018, 9:01 pm

All was not lost; at the time Schwartz, Sobolev and others took the delta function by the scruff of the neck and generalize to the theory of Distributions, without which there would be no Finite Element Method.

The concrete example again..

So, instead of huffing and puffing with Fourier and integral, just go for the Cauchy distribution?
Throwing Cauchy'n to the wind, what?
Cauchy me here and there. Can the conclusion be explained also to a mortal like me? 
 
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Cuchulainn
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Re: Philosophy of Mathematics

May 26th, 2018, 12:26 pm

 
Cauchy has more theorems named after him than anyone else (he even invented the neural networkers' gradient descent; some folk believe GD was invented at the MIT AI Lab..).
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