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jhooper3581
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Newton FTW
Who divided by zero numerous times to invent it?hamster143 said:Einstein wasn't particularly good in mathematics, he had to pick Hilbert's brain for a lot of GR-related stuff.
And you're trying to pit him against the guy who invented calculus.
hamster143 said:And you're trying to pit him against the guy who invented calculus.
To mathematics?BobG said:How big of a coincidence is it that he and Leibniz invented calculus at approximately the same time?
Maybe it's not. Maybe calculus was just a natural progression from DesCarte's analytic geometry.
In any event, I still think Newton made more fundamental contributions than Einstein.
Kajahtava said:calculus can hardly be called fundamental
Really now? where does one encounter calculus (more properly called infinitesimal calculus) in:Jack21222 said:Err, what? Calculus is a fundamental part of almost all math above high school algebra.
I do not think it is very wise to display such strong opinions towards what is "more fundamental". The interplay between analysis and algebra remains at all stages of mathematical sophistication. Consider linear algebra : a lot of Hilbert space constructions were motivated by harmonic analysis. Numerous theorems in number theory are obtained using complex analysis. In fact, I'll just quote Riemann's hypothesis : from the definition of the hypothesis to the latest bright idea to try to prove it through the entire history of the problem, we keep going back and forth between analysis and algebra.Kajahtava said:Infinitesimal calculus is simply a useful tool that can be used to calculate some magnitudes, there is no fundamental research going on in it.
Fundamental is easy to define. You can express / formulate calculus in set theory, but not the reverse; thus set theory is more fundamental.humanino said:I do not think it is very wise to display such strong opinions towards what is "more fundamental".
Algebra isn't exactly fundamental either. Algebra uses numbers and the operations thereon and accepts them as existing axioms, fundamental mathematics is more interested in first defining what a number is in a given context, what a certain operation on numbers is.The interplay between analysis and algebra remains at all stages of mathematical sophistication.
Riemann Hypothesis isn't as much fundamental as it is far-reaching. An example of a fundamental hypothesis would be the Church-Turing thesis.Consider linear algebra : a lot of Hilbert space constructions were motivated by harmonic analysis. Numerous theorems in number theory are obtained using complex analysis. In fact, I'll just quote Riemann's hypothesis : from the definition of the hypothesis to the latest bright idea to try to prove it through the entire history of the problem, we keep going back and forth between analysis and algebra.
Brilliant maybe, but mathematically brilliant hardly.DaveC426913 said:Well, more fundamental or not: in the context of the OP's question, which is about mathematical brilliance, Newton was pretty mathematically brilliant to invent calculus.
Kajahtava said:Brilliant maybe, but mathematically brilliant hardly.
Calculus is mathematically dubious, it just had profoundly wide application and use, but a work of mathematics it's not. It's basically just a trick, the larger trick is to disguise the fact that you divide by zero.
Why is it forbidden you might ask yourself?DaveC426913 said:Why do you minimize that as a "trick"? We can't divide by zero because it's forbidden, but the success of calculus comes from the fact that it is very often very useful to do so.
No, I'm just saying that calculus how Newton invented it is not mathematics.Are you going to split hairs and suggest that it's not enough to invent something spectacularly useful and succesful?
I am not using the terms "analysis" and "algebra" as specific branches which for instance could be taught in school. I am referring to a more general split of all mathematical concepts.Kajahtava said:Algebra uses numbers and the operations thereon and accepts them as existing axioms, fundamental mathematics is more interested in first defining what a number is in a given context, what a certain operation on numbers is.
Kajahtava said:Why is it forbidden you might ask yourself?
How is calculus not mathematics? It's like saying the invention of the mirror is not about optics.Kajahtava said:No, I'm just saying that calculus how Newton invented it is not mathematics.
The invention of the mirror was also highly useful, does that make it mathematically brilliant? Of course not, though one could argue that it was brilliant on its own.
I believe it to be quite relevant.DaveC426913 said:No, I did not ask myself that. The rest of what you said is irrelevant, but thanks for sharing.
Well, I doubt the person that invented the mirror knew any thing about optics, in fact, I think it for the most part was just dumb luck to be honest.How is calculus not mathematics? It's like saying the invention of the mirror is not about optics.
What split is that? You mean there is some binary (or higher) split between all branches of mathematics? I fail to understand what you mean.humanino said:I am not using the terms "analysis" and "algebra" as specific branches which for instance could be taught in school. I am referring to a more general split of all mathematical concepts.
You had to put words in my mouth to justify explaining it. Everyone knows why dividing by zero is forbidden. It doesn't say anything about calculus.Kajahtava said:I believe it to be quite relevant.
Ah, that's the answer.Kajahtava said:It's not mathematics for the same reason that 'proving' the Riemann Hypothesis by saying 'Okay, we found a thousand cases where it applies no and no counter example, it then must be true', is not mathematics, it may be useful, and this is how most empirical sciences work, but it's not how mathematics works.
Analysis and algebra. All mathematicians are familiar with this split, as it corresponds to real occurring preferences among professionals. I no of no mathematician who would claim their preference to be "superior" or more fundamental to the other one. One can take the list of Field medalists and classify the work accordingly. I'm pretty sure Perelman's work for which he was attributed the Field medal would fall in the "analysis" category for instance, although I do not know him so I do not know his personal preference. So would Tao's Field medal work, or Wendelin Werner Field medal work, or René Thom's Field medal work.Kajahtava said:What split is that?
Are you a published researcher in mathematics ?Kajahtava said:it's not how mathematics works.
What? This is new to me?humanino said:Analysis and algebra. All mathematicians are familiar with this split, as it corresponds to real occurring preferences among professionals. I no of no mathematician who would claim their preference to be "superior" or more fundamental to the other one. One can take the list of Field medalists and classify the work accordingly. I'm pretty sure Perelman's work for which he was attributed the Field medal would fall in the "analysis" category for instance, although I do not know him so I do not know his personal preference. So would Tao's Field medal work, or Wendelin Werner Field medal work, or René Thom's Field medal work.
The reason I am quoting Field medal work falling in the category of analysis, is that my own preference is algebra.
No, do you need one to back up my claim that in mathematics, finding a thousand positive examples to the Riemann Hypothesis and no negative example is enough a substantiation for the claim?humanino said:Are you a published researcher in mathematics ?
Hmm, I don't know about you, but when I still attended university saying 'it's not mathematical' was essentially the same thing as saying 'it's not rigorous'. I would call it 'a useful trick' opposed to mathematics if it lacks rigour.DaveC426913 said:Ah, that's the answer.
You're not saying it's not mathematics, you're saying is not rigorous.
You could have been a little more forthright.
Kajahtava said:Brilliant maybe, but mathematically brilliant hardly.
Calculus is mathematically dubious, it just had profoundly wide application and use, but a work of mathematics it's not. It's basically just a trick, the larger trick is to disguise the fact that you divide by zero.
M Grandin said:What do you mean by "divide by zero"? Is it the quotient dy/dx , where dx is approaching zero? It is the limit of dy/dx when dx approaches zero, that is meant. If for instance a line y = kx + l , then that quotient dy/dx = k however small you make dx. That must be easy realize.
Both are algebraic.Kajahtava said:what does proof theory fall into? Algebra? What does linear algebra fall into?
Topology is not easy to classify in algebra or analysis, it could be on both side. Algebraic topology is definitely not analysis for instance.Kajahtava said:I take it that if topology falls into analysis by this schism that linear algebra also falls in analysis?
Hmm, interesting, I at first mistakenly was inclined to perceive 'more formal' as 'algebraic' from your perception.humanino said:Both are algebraic.
Topology is not easy to classify in algebra or analysis, it could be on both side. Algebraic topology is definitely not analysis for instance.
This is not a well-defined classification, but if you work with mathematicians, the majority of them have a sensitivity towards one or the other side.
Kajahtava said:Why is it forbidden you might ask yourself?
Because zero has no multiplicative inverse, after all, division by x is defined as multiplying by the multiplicative inverse of x.
The multiplicative inverse of a real number x is a number y such that x multiplied by y results into 1.
It is provably that each and every real number has exactly one such multiplicative inverse, except 0, and no real number has 0 as multiplicative inverse. Because of course the inverse of the inverse is the number itself, a thing that's also provable.
As you said, it is very useful, it's also very useful to treat pi as 3.14 in most circumstances, because the result, though only an approximation, is close enough to what we need, though doing so is where you stop performing mathematics.
No, I'm just saying that calculus how Newton invented it is not mathematics.
The invention of the mirror was also highly useful, does that make it mathematically brilliant? Of course not, though one could argue that it was brilliant on its own.
What do you mean with 'fundamental to day to day life'?Jack21222 said:You have a different definition of the word "mathematics" than most of the world does. You're also using a different definition of the word "fundamental."
Tell me, what's more fundamental to day to day life? Calculus, or topology?
Jack21222 said:...what's more fundamental to day to day life...
Maybe it would be best if I admitted I would rather classify them as "foundational mathematics". Since if I had to classify them either as "algebra" or "analysis", I would pick algebra, it is possible that we actually share the same point of view.Kajahtava said:But what is proof theory then, or lambda calculus, or set theory, or recursion theory?
Hmm, maybe this is your own environment, most I know are foundational mathematicians, but then again, that could be my own environment.humanino said:Maybe it would be best if I admitted I would rather classify them as "foundational mathematics". Since if I had to classify them either as "algebra" or "analysis", I would pick algebra, it is possible that we actually share the same point of view.
Maybe where we agree the least is that, when I say "foundational mathematics", I really mean that so few mathematicians actually work on "foundational mathematics", they are really on the fringe to me. But you may also call it the Heart, with a capital. The classification "algebra" vs "analysis" is just a gross feature to describe roughly in the vast world of mathematics. It is not a classification which is really relevant to the very specific field of "foundational mathematics", since it is already quite restricted and much better defined than "algebra" vs "analysis".