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Cuchulainn
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Re: Philosophy of Mathematics

October 4th, 2020, 1:47 pm

The modern foundations of mathematical analysis were established in 17th century Europe.[3] Descartes and Fermat independently developed analytic geometry, and a few decades later Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variationsordinary and partial differential equationsFourier analysis, and generating functionsDuring this period, calculus techniques were applied to approximate discrete problems by continuous ones.

So that's how those old-timers got all those magic ODEs and PDEs? Crafty devils!
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Re: Philosophy of Mathematics

October 5th, 2020, 10:52 pm

who first discovered/derived the advection equation?
 
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Re: Philosophy of Mathematics

October 10th, 2020, 3:23 pm

who first discovered/derived the advection equation?
Riemann? (geometry)
Last edited by Cuchulainn on October 10th, 2020, 3:26 pm, edited 1 time in total.
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Re: Philosophy of Mathematics

October 10th, 2020, 3:24 pm

When did it start to go wrong very badly with maths and physics? (beginning of 20th century, beginning of the end).

www.youtube.com/watch?v=avSHHi9QCjA

CL once suggested to me to generalise the Radon-Nikodym theorerm to the complex case. Of course, it was proved earlier but you don't know that at the time.
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Re: Philosophy of Mathematics

October 10th, 2020, 10:50 pm

I think biology took over the scientific narration around mid 20th century. It outsourced all creative and talented people. Physics was commercialised to efficiently reap the benefits of its predecessors and colonised by con artists who repaint it into a increasingly caricatural picture.
 
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Re: Philosophy of Mathematics

October 11th, 2020, 11:35 am

And String Theory hype reminded me of snake oil.
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Re: Philosophy of Mathematics

October 11th, 2020, 6:45 pm

Gil Strang on teaching Linear Algebra

www.youtube.com/watch?v=7UJ4CFRGd-U
Excellent! I agree 100%. The only thing I’d suggest is that he uses more politically incorrect jokes and analogies in his lectures.
He probably has no time for chit-chat.

www.youtube.com/watch?v=sx00s7nYmRM
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Re: Philosophy of Mathematics

October 15th, 2020, 10:36 pm

wannabes, your last hope is out: Rejecta Mathematica
 
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Re: Philosophy of Mathematics

October 15th, 2020, 11:00 pm

wannabes, your last hope is out: Rejecta Mathematica
Commies, all of them
Met Wikisage hopen we het beter te doen door zowel de encyclopedie als de gemeenschap een andere structuur te geven. Nieuw is, om te beginnen:
  • aan een encyclopedisch artikel kunnen essays worden toegevoegd van individuele gebruikers
  • een expert kan worden benoemd tot (wiki-)Wijze ('Sage') die de verantwoordelijkheid heeft voor een of meer artikelen
Een noviteit is ook het begrip (wiki-)dorp. Gebruikers die op hetzelfde terrein bezig zijn kunnen een dorp stichten. Ingeval van een inhoudelijk conflict op hun terrein geeft de mening van het dorp de doorslag. Dit nodigt uit tot samenwerking en biedt de mogelijkheid van kleinschalige sociale controle.
Kortom, op Wikisage is er respect voor elkaars deskundigheid. Zo is ook het refereren aan eigen publicaties een normaal verschijnsel, indien deze publicaties verifieerbaar en relevant zijn.
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Re: Philosophy of Mathematics

October 16th, 2020, 8:28 am

I searched this long post to find some mention of Frege 

https://plato.stanford.edu/entries/frege/

and surprised to find none, given his name is the one most associated with 'philosophy of mathematics'.

Briefly, Frege sought to ground mathematics in logic, which he set out in his famous Grundgesetze der Arithmetik. Russell pointed out that one could derive a contradiction in the system in terms of the class of all those classes that are not members of themselves.  Is it a member of itself or not. 

"Frege never fully recovered from the fatal flaw discovered in the foundations of his Grundgesetze." 

Zermelo was more successful in axiomatising set theory.
 
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Re: Philosophy of Mathematics

October 18th, 2020, 12:08 pm

Why the surprise? The word "p(P)hilosophy" is context-sensitive and it can have various connotations. Not everyone knows or is interested in Frege I suppose.

Mathematics was in crisis at the end of the 19th century. Frege's Begrifsschrift was a failed attempt to define a formal language for arithmetic but the real breakthrough was by Richard Dedekind (Gauss last Phd student) when he invented cuts (it's in Rudin's book). Actually, cuts were a "filtering" mechanism for 1st year undergrad maths students. You got it or you didn't..

Frege's theory of Concept and Object also came in for a lot of criticism. Unfortunately, it influenced the foundations of Object Oriented as it was too rigid ("all mammals are land-dwellers"). Many developers are designing and maintaining semantically incorrect class libraries.

I think Frege influenced Ayn Rand's Objectivist Epistemology but the less we say about that the better.

“Philosophy is a battle against the bewitchment of our intelligence by means of language.”

― Ludwig Wittgenstein, Philosophical Investigations
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Re: Philosophy of Mathematics

October 18th, 2020, 5:49 pm

Why the surprise? The word "p(P)hilosophy" is context-sensitive and it can have various connotations. Not everyone knows or is interested in Frege I suppose.

Mathematics was in crisis at the end of the 19th century. Frege's Begrifsschrift was a failed attempt to define a formal language for arithmetic but the real breakthrough was by Richard Dedekind (Gauss last Phd student) when he invented cuts (it's in Rudin's book). Actually, cuts were a "filtering" mechanism for 1st year undergrad maths students. You got it or you didn't..

Frege's theory of Concept and Object also came in for a lot of criticism. Unfortunately, it influenced the foundations of Object Oriented as it was too rigid ("all mammals are land-dwellers"). Many developers are designing and maintaining semantically incorrect class libraries.

I think Frege influenced Ayn Rand's Objectivist Epistemology but the less we say about that the better.

“Philosophy is a battle against the bewitchment of our intelligence by means of language.”

― Ludwig Wittgenstein, Philosophical Investigations
Not sure about any of this, tbh. "Not everyone knows or is interested in Frege" - Frege was the biggest single influence on philosophy of language in the mid to late 20C. He had less influence on the development of mathematics. 

"I think Frege influenced Ayn Rand's Objectivist Epistemology but the less we say about that the better." Professional philosophers mostly ignore Rand. 
 
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Re: Philosophy of Mathematics

October 18th, 2020, 6:35 pm

We have moved to "philosophy of language" now? That's quite a jump from "philosophy of maths"

 "Not everyone knows or is interested in Frege"
I was referring to the good folk on the Wilmott forum, I have no idea how Frege is perceived on the outside.
Maybe I can study Frege after my retirement.

Professional philosophers mostly ignore Rand. 
Half of the US Republican Party swear by Ayn Rand.
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Re: Philosophy of Mathematics

October 18th, 2020, 8:36 pm

We have moved to "philosophy of language" now? That's quite a jump from "philosophy of maths"
Not at all. Frege introduced the idea of the propositional function, thus taking an idea that is fundamental to mathematics into logic and semantics. Thus 'is a philosopher' is a function taking individuals as arguments, and mapping them to truth values. e.g is-a-philosopher(Russell) = True, is-a-philosopher(Trump)  = False.

Long story though.
 
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Re: Philosophy of Mathematics

October 18th, 2020, 10:03 pm

We have moved to "philosophy of language" now? That's quite a jump from "philosophy of maths"
Not at all. Frege introduced the idea of the propositional function, thus taking an idea that is fundamental to mathematics into logic and semantics. Thus 'is a philosopher' is a function taking individuals as arguments, and mapping them to truth values. e.g is-a-philosopher(Russell) = True, is-a-philosopher(Trump)  = False.

Long story though.
I can handle it; have a try.
It's long but can be summarised Frege's defining attribute view of concepts. The intension of a concept is the set of attributes that defines who it is to be a member of the concept and the extension is the set of entities that are members of the concept. This is class-based object oriented paradigm.

Saying   is-a-philosopher(Trump)  = False. is typical Frege but wrong because real-life is context-sensitive. After losing in a few weeks he could decide to do philosophy at Fordham then  is-a-philosopher(Trump)  = True. And a degree in faucet-making at MIT and  is-a-plumber(Trump)  = True. But that would be multiple classification! Ouch.

The C++ database would have to be recalibrated. Frege missed out on cognitive psychology.

Prolog is programming language based on Frege logic. It never caught on. (many burnt fingers in the past).
Last edited by Cuchulainn on October 18th, 2020, 10:32 pm, edited 2 times in total.
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