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exneratunrisk
Posts: 3559
Joined: April 20th, 2004, 12:25 pm

### Philosophy of Mathematics

QuoteOriginally posted by: And2QuoteOriginally posted by: exneratunrisk the "world" in which mathematics can act as "quantifier eliminator" is small, the pragmatists (number crunchers) took over. "For all" means now (for all finite cases, I have tested ..).Exneratunrisk,This does not make sense, especially after Goedel...Anyway, here is another good reference, not exactly about mathematics, but has a lot of that too.Cuchulainn,This is great video. Thank you for posting.Especially GEB shows that thinking happens on an interaction layer between the formal and the semantic system. This does not speak against maths, but it gives us an idea that reductionism is not everything. The philosophers of speculative realism use a great trick: if we do not know more properties of a real behavior than those in our models, then our models ARE reality (a kind of Black Scholes trick )p.s. Gödel: all systems more complex than arithmetic have undecidable expressions.

Cuchulainn
Posts: 62920
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Philosophy of Mathematics

QuoteEspecially GEB shows that thinking happens on an interaction layer between the formal and the semantic systemI tried reading that stuff but somehow it did not turn me on. Mixing maths and alchemy never works.Have a look at Dedekind cuts which our Prof. at TCD used to cull the number of 1st year maths students from 50 to 10. A mouthfulQuoteaxiom: If A is a non-empty subset of R, and if A has an upper bound, then A has a least upper bound u, such that for every upper bound v of A, u ≤ v.The final axiom, defining the order as Dedekind-complete, is most crucial. Without this axiom, we simply have the axioms which define a totally ordered field, and there are many non-isomorphic models which satisfy these axioms. This axiom implies that the Archimedean property applies for this field. Therefore, when the completeness axiom is added, it can be proved that any two models must be isomorphic, and so in this sense, there is only one complete ordered Archimedean field.And this gives us real numbers.
Last edited by Cuchulainn on September 17th, 2013, 10:00 pm, edited 1 time in total.
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And2
Posts: 103
Joined: January 29th, 2007, 5:24 pm

### Philosophy of Mathematics

QuoteOriginally posted by: Cuchulainn... it can be proved that any two models must be isomorphic, and so in this sense, there is only one complete ordered Archimedean field.This kind of theorems always fascinated me. You start with few constrains, allowing one to do/have almost anything, and suddenly your world is shrank to very few possible structures.A few more examples that gave me (still do) religious experience: 1. Any complete metric field is either Archimedean (isomorphic to R or C), or one from countable set of non-Archimedean fields (isomorphic to p-adic fields) - basically, there are only so much different norms (or metrics), and only two different types (Archimedean, and p-adic). 2. There are only so much differnet types of (finite simple) groups - especially fascinating are their sporadic groups3. CLT - or there is only one Gaussian distribution
Last edited by And2 on September 17th, 2013, 10:00 pm, edited 1 time in total.

ThinkDifferent
Posts: 659
Joined: March 14th, 2007, 1:09 pm

### Philosophy of Mathematics

QuoteOriginally posted by: And2QuoteOriginally posted by: CuchulainnA few more examples that gave me (still do) religious experience: 1. Any complete metric field is either Archimedean (isomorphic to R or C), or one from countable set of non-Archimedean fields (isomorphic to p-adic fields) - basically, there are only so much different norms (or metrics), and only two different types (Archimedean, and p-adic). 2. There are only so much differnet types of (finite simple) groups - especially fascinating are their sporadic groups3. CLT - or there is only one Gaussian distributionI think this 'religious' experience comes from ignorance. Just like with a real religion i.e. the less u know about the subject the more religious is your experience coming from algebraic background i do not find 1 and 2 as very striking.I'd somewhat agree about 3. Again, as a non-geometrist, I find something like Banah-Tarski paradox or classification of exotic spheres quite fascinating.

Ultraviolet
Posts: 1655
Joined: August 15th, 2012, 9:46 am

### Philosophy of Mathematics

Banach

Cuchulainn
Posts: 62920
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Philosophy of Mathematics

Quote Again, as a non-geometrist, I find something like Banah-Tarski paradox or classification of exotic spheres quite fascinating. Geometry and arithmetic. And Cauchy sequences! typo (see UV remark) You attribute And2's 'religious' quote to me.LOL
Last edited by Cuchulainn on September 18th, 2013, 10:00 pm, edited 1 time in total.
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ThinkDifferent
Posts: 659
Joined: March 14th, 2007, 1:09 pm

### Philosophy of Mathematics

QuoteOriginally posted by: CuchulainnQuote Again, as a non-geometrist, I find something like Banah-Tarski paradox or classification of exotic spheres quite fascinating. Geometry and arithmetic. And Cauchy sequences! typo (see UV remark) You attribute And2's 'religious' quote to me.LOLah, sorry. not sure how that happened. I used a 'Quote' button. and thanks UV for a deep remark

Cuchulainn
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### Philosophy of Mathematics

QuoteMathematics is the art of giving the same name to different thingsHenri Poincaré QuoteMost scholars are excellent problem solvers, but have spent little time understandinghow they have reached their solutions ? computer scientists are no exception. Little ofthe published information studied in this paper about the computer scientists selectedoffered a different perspective of how psychology and computer science can correlateand produce creative results. This, of course, may be due to the very limited numberof writings reviewed for this paper. However, the difficulty may lie deeper. Thewritings excerpted from Out of Their Minds attempt to detail the ideas of the subjectsby recalling situations and instances, as well as providing motivation for their work.Contrast this with the writings of Poincaré and Hadamard. Poincaré discussedillumination and discernment. Although, his view might be limited based upon thepremise that one cannot analyze each thought while having that thought..
Last edited by Cuchulainn on September 18th, 2013, 10:00 pm, edited 1 time in total.
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And2
Posts: 103
Joined: January 29th, 2007, 5:24 pm

### Philosophy of Mathematics

Last edited by And2 on September 18th, 2013, 10:00 pm, edited 1 time in total.

Cuchulainn
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Joined: July 16th, 2004, 7:38 am
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### Philosophy of Mathematics

Last edited by Cuchulainn on September 18th, 2013, 10:00 pm, edited 1 time in total.
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Cuchulainn
Posts: 62920
Joined: July 16th, 2004, 7:38 am
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### Philosophy of Mathematics

Quote... I have grown very stupid of late, and regularly fail at everything I attempt. What the reason may be I cannot tell. But I begin to be of Newton's opinion, that after a certain age, a man may as well give up mathematics. ... James McCullough 1847
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Cuchulainn
Posts: 62920
Joined: July 16th, 2004, 7:38 am
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### Philosophy of Mathematics

When learning new stuff, feels likeQuoteIn mathematics you don't understand things. You just get used to them. John von Neumann
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Cuchulainn
Posts: 62920
Joined: July 16th, 2004, 7:38 am
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### Philosophy of Mathematics

QuoteThis new integral of Lebesgue is proving itself a wonderful tool. I might compare it with a modern Krupp gun, so easily does it penetrate barriers which were impregnable.Edward Van Vleck, Bulletin of the American Mathematical Society, vol. 23, 1916.
Last edited by Cuchulainn on November 14th, 2013, 11:00 pm, edited 1 time in total.
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Cuchulainn
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Joined: July 16th, 2004, 7:38 am
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### Philosophy of Mathematics

QuoteThe following is a list of some of the criticisms commonly made of the Bourbaki approach. Pierre Cartier, a Bourbaki member 1955?1983, commented explicitly on several of these points:[10] ...essentially no analysis beyond the foundations: nothing about partial differential equations, nothing about probability. There is also nothing about combinatorics, nothing about algebraic topology, nothing about concrete geometry. And Bourbaki never seriously considered logic. Dieudonné himself was very vocal against logic. Anything connected with mathematical physics is totally absent from Bourbaki's text.
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Cuchulainn
Posts: 62920
Joined: July 16th, 2004, 7:38 am
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### Philosophy of Mathematics

QuoteMentally challenged zealots of "abstract mathematics" threw all the geometry (through which connection with physics and reality most often takes place in mathematics) out of teaching. Calculus textbooks by Goursat, Hermite, Picard were recently dumped by the student library of the Universities Paris 6 and 7 (Jussieu) as obsolete and, therefore, harmful (they were only rescued by my intervention). V.I. Arnold
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