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katastrofa
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Re: Philosophy of Mathematics

October 1st, 2018, 9:33 pm

So was mine ;-) (at scientific institutes with true scientists). I'm surprised to learn that you read quantum mechanics considering some conversations from the forum. Must have been long ago.

Seriously, I assisted to mathematical analysis lectures at a London uni as a postdoc and it was pretty basic. Nothing my courses in the physics curriculum didn't cover during the first 3 years.
Last edited by katastrofa on October 2nd, 2018, 10:35 am, edited 1 time in total.
 
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katastrofa
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Re: Philosophy of Mathematics

October 1st, 2018, 11:21 pm

To be fair to Cuch there is no substitute for an undergrad maths degree. Not physics, or electrical engineering or economics or, lord help us, finance.

(But the friend of a friend osmosis thing was rather weak!)
Especially for someone knowledgeable about things like the Riemann hypothesis.
 
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Paul
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Re: Philosophy of Mathematics

October 3rd, 2018, 8:41 am

... his colleagues no longer talk to him... 
His colleagues have always called him "The Ayatollah."
 
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Cuchulainn
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Re: Philosophy of Mathematics

November 7th, 2018, 4:11 pm

In his book Psychology of Invention in the Mathematical Field,[6] Hadamard uses introspection to describe mathematical thought processes. In sharp contrast to authors who identify language and cognition, he describes his own mathematical thinking as largely wordless, often accompanied by mental images that represent the entire solution to a problem. He surveyed 100 of the leading physicists of the day (approximately 1900), asking them how they did their work.
 
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Paul
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Re: Philosophy of Mathematics

November 7th, 2018, 4:46 pm

Interesting. Pure mathematicians versus applied? I don’t use language much in formulating a problem/solution, at least initially. I tend to imagine being “inside” a problem, almost physically moving things around and seeing how they react. I do sometimes dream of an approach to a problem and often it turns out to work.
 
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katastrofa
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Re: Philosophy of Mathematics

November 7th, 2018, 6:04 pm

I'm struggling to imagine physicists thinking about mathematics, because most of those I've met didn't know it.
 
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Cuchulainn
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Re: Philosophy of Mathematics

November 10th, 2018, 4:50 pm

I'm struggling to imagine physicists thinking about mathematics, because most of those I've met didn't know it.
I think it is more nuanced.

No two languages are ever sufficiently similar to be considered as representing the same social reality. The worlds in which different societies live are distinct worlds, not merely the same world with different labels attached. Edward Sapir
..
The Sapir-Whorf hypothesis proclaimed the influence of language on thought and perception. This, in turn, implies that the speakers of different languages think and perceive reality in different ways and that each language has its own world view. The issues this hypothesis raised not only pertain to the field of linguistics but also had a bearing on Psychology, Ethnology, Anthropology, Sociology, Philosophy, as well as on the natural sciences. For, if reality is perceived and structured by the language we speak, the existence of an objective world becomes questionable, and the scientific knowledge we may obtain is bound to be subjective. Such a principle of relativity then becomes a principle of determinism. Whether the language we speak totally determines our attitude towards reality or whether we are merely influenced by its inherent world view remains a topic of heated discussion.

A nice mini project would be to take any (extended) discussion on let's say Numerical/Programming and to analyse how mathematicians, physicists,. CS types, economists and engineers differ based solely on how language has influenced their thought processes.

There have been many such epiphanies  down the years. My favourites are 1) negative probabilities, 2) A la-carte definition of exp(Matrix), 3) Cauchy sequences are only in the mind of mathematicians, 4) links to ML working articles.

A lot of science is description, not explanation. In a sense, kind of useless.
 
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katastrofa
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Re: Philosophy of Mathematics

November 11th, 2018, 3:41 am

Negative probabilities are still alive and used by ignorant physicists to explain phenomena as "quantum mechanical". I would add to your list the popular interpretation of quantum mechanics itself and e.g. the no-free-lunch theory.

You've probably read about Himba tribe from Namibia. They have a different system of colours. It reflects their natural environment (e.g. they distinguish over 10 types of green as they see them in local vegetation). Imagine that each colour represents some area or discipline of science. Those who study this discipline have a tendency to focus on what's within its limits, like colour blue is about 600-680 THz or green is 520-600THz. If we change the scope of the discipline, we will focus on different "frequencies" and have the possibility to look at it from a new perspective, see new relationships between subjects, new interpretations, etc. For Himba blue and some green colours have the same name.

Image

On the other hand, I don't verbalise my thoughts (communicating them is a significant effort). Do you?
 
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rmax
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Re: Philosophy of Mathematics

November 12th, 2018, 3:24 pm

In Welsh Green and Blue are the same colour.
 
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rmax
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Re: Philosophy of Mathematics

November 12th, 2018, 3:25 pm

Image
 
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katastrofa
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Re: Philosophy of Mathematics

November 12th, 2018, 4:00 pm

Inuit?
 
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Cuchulainn
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Re: Philosophy of Mathematics

November 12th, 2018, 4:48 pm

Here's a question on how to overcome the contradiction and incompatibility between Fourier's equation (Fick's Law) and the theory of relativity (in the latter case speed of heat propagation is infinite while in the latter case nothing is faster than light).

A fix for this conundrum is to stick a 2nd order time derivative to Fourier PDE and hey presto resulting in the Hyperbolic Heat Conduction (HHC) equation, a bit like the telegrapher's equation. But we need to introduce the second sound into the PDE.

IMO it is the same tale as with negative probabilities.

Questions

1. This is not the mathematical approach ... too many assumptions (in fact, it violates the 2nd law of thermodynamics).
2. Second sound is fiction (it has no physical reality)
3. HHC is a weak description of relativity but it is still Newtonian.
4. You can't just stick bits and pieces to a PDE and hope it will work. (disclaimer: I did QM/SR/GR at undergrad but I don't remember 1/2 of it)

At which step in inventing HHC did it go wrong?  What is the major (implicit) assumption here?

Image
 
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Cuchulainn
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Re: Philosophy of Mathematics

November 18th, 2018, 12:15 pm

Image
 
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Cuchulainn
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Re: Philosophy of Mathematics

November 19th, 2018, 10:43 am

Maths teacher; who needs trigonometry, anyways?
 
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katastrofa
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Re: Philosophy of Mathematics

November 20th, 2018, 7:20 am

Here's a question on how to overcome the contradiction and incompatibility between Fourier's equation (Fick's Law) and the theory of relativity (in the latter case speed of heat propagation is infinite while in the latter case nothing is faster than light).

A fix for this conundrum is to stick a 2nd order time derivative to Fourier PDE and hey presto resulting in the Hyperbolic Heat Conduction (HHC) equation, a bit like the telegrapher's equation. But we need to introduce the second sound into the PDE.

IMO it is the same tale as with negative probabilities.

Questions

1. This is not the mathematical approach ... too many assumptions (in fact, it violates the 2nd law of thermodynamics).
2. Second sound is fiction (it has no physical reality)
3. HHC is a weak description of relativity but it is still Newtonian.
4. You can't just stick bits and pieces to a PDE and hope it will work. (disclaimer: I did QM/SR/GR at undergrad but I don't remember 1/2 of it)

At which step in inventing HHC did it go wrong?  What is the major (implicit) assumption here?

Image
IMHO, HHC is just a crude approximation, while negative probabilities are simply an erroneously used name for signed measures or Wigner function (which is a quasi-probability). Another depressing (with its prevalence) curiosity is - assuming that we've lost the battle for the "quasi" prefix - claiming that the negative probability (negative values of the Wigner function) indicates quantum behaviour of a system. In fact, there are multum of examples of quantum systems which have non-negative everywhere Wigner functions and classical systems with negative Wigner functions in student textbooks). Who cares, though - nowadays scientists need to produce a lot of buzz and Nature publications, and not something actually useful.
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