Who Was the Ultimate Genius: Newton or Riemann?

[jhooper3581]

Newton FTW

 

[humanino]

Archimedes.

 

[Cyrus]

Humanino.

 

[KrisOhn]

I actually agree with humanino, but out of the two given I'd have to go with Newton.

 

[hamster143]

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.

 

[Kajahtava]

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.

Who divided by zero numerous times to invent it?

Both produced some 'tricks' which had tremendous use. But Einstein's work wasn't that endeavouring, and Newton's work was simply inconsistent, he divided by zero, come on.

 

[BobG]

And you're trying to pit him against the guy who invented calculus.

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.

 

[Phrak]

My vote goes to Forest, Forest Gump.

He invented the happy face. He was so renouned, 24 US presidents shook his hand and LBJ even examined his but tox.

 

[Kajahtava]

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.

To mathematics?

Einstein didn't make any contribution to mathematics as far as I know, calculus can hardly be called fundamental. Both made some very fundamental contributions to physics though.

 

[Jack21222]

calculus can hardly be called fundamental

Err, what? Calculus is a fundamental part of almost all math above high school algebra.

 

[Kajahtava]

Err, what? Calculus is a fundamental part of almost all math above high school algebra.

Really now? where does one encounter calculus (more properly called infinitesimal calculus) in:

- cryptography
- linear algebra
- proof theory
- set theory
- propositional logic
- first order logic
- functional analysis
- topology
- number theory

Calculus is probably the least fundamental part of maths out there. Calculus is founded on analysis, analysis is founded on topology, topology is founded on set theory, set theory is founded on first order logic, itself founded on proof theory. Nothing I know of in mathematics is founded on infinitesimal calculus, but maybe you can teach me some new things here.

Infinitesimal calculus is simply a useful tool that can be used to calculate some magnitudes, there is no fundamental research going on in it.

 

[humanino]

Infinitesimal calculus is simply a useful tool that can be used to calculate some magnitudes, there is no fundamental research going on in it.

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]

I do not think it is very wise to display such strong opinions towards what is "more fundamental".

Fundamental is easy to define. You can express / formulate calculus in set theory, but not the reverse; thus set theory is more fundamental.

The interplay between analysis and algebra remains at all stages of mathematical sophistication.

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.

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.

Riemann Hypothesis isn't as much fundamental as it is far-reaching. An example of a fundamental hypothesis would be the Church-Turing thesis.

 

[DaveC426913]

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]

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.

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.

 

[DaveC426913]

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 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.

Are you going to split hairs and suggest that it's not enough to invent something spectacularly useful and succesful?

 

[Kajahtava]

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.

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.

Are you going to split hairs and suggest that it's not enough to invent something spectacularly useful and succesful?

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.

 

[humanino]

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.

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.

 

[DaveC426913]

Why is it forbidden you might ask yourself?

No, I did not ask myself that. The rest of what you said is irrelevant, but thanks for sharing.

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.

How is calculus not mathematics? It's like saying the invention of the mirror is not about optics.

 

[Kajahtava]

No, I did not ask myself that. The rest of what you said is irrelevant, but thanks for sharing.

I believe it to be quite relevant.

How is calculus not mathematics? It's like saying the invention of the mirror is not about optics.

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.

Newton's Calculus is not mathematics because one divides by zero. Or at least, he offered little explanation to the existence of an object dx which we can add to a number r to produce r again. (therefore it must be zero, or we must define addition on some larger set) and then divides by it randomly as if it's not zero.

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.

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.

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.

What I mean is that calculus is not mathematics, analysis is mathematics, but not fundamental mathematics.

 

[DaveC426913]

I believe it to be quite relevant.

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.

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.

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.

 

[humanino]

What split is that?

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.

 

[humanino]

it's not how mathematics works.

Are you a published researcher in mathematics ?

 

[Kajahtava]

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.

What? This is new to me?

So ehh, what does proof theory fall into? Algebra? What does linear algebra fall into?

I take it that if topology falls into analysis by this schism that linear algebra also falls in analysis?

Are you a published researcher in mathematics ?

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?

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.

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.

 

[M Grandin]

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.

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.

 

[BobG]

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.

Limits were developed later by Cauchy. Newton had no limits.

 

[humanino]

what does proof theory fall into? Algebra? What does linear algebra fall into?

Both are algebraic.

I take it that if topology falls into analysis by this schism that linear algebra also falls in analysis?

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]

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.

Hmm, interesting, I at first mistakenly was inclined to perceive 'more formal' as 'algebraic' from your perception.

But what is proof theory then, or lambda calculus, or set theory, or recursion theory?

 

[Jack21222]

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.

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?

 

[Kajahtava]

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?

What do you mean with 'fundamental to day to day life'?

http://en.wikipedia.org/wiki/Foundational_mathematics

Fundament / foundation: that on which other things are built.

Every day life is built in calculus, calculus is built on analysis, analysis is built on topology, topology on set theory, set theory on first order logic, that on proof theory.

Though we can argue that first order logic and set theory are aequally fundamental, as first order logic can also be expressed in set theory. Set theory however cannot be expressed in topology, but the reverse can, thus we say that set theory is more fundamental than topology. I would hardly find this nonstandard a definition, and seeing that wikipedia still collects the opiate of the masses, its agreeing with me seems to indicate that it's the standard definition.

 

[DaveC426913]

...what's more fundamental to day to day life...

You've taken the word out of context. It was applying to mathematics; you're applying it to daily life.

And you're using a definition of fundamental that is synonymous with 'important', as opposed to a defintion of fundamental that is synonymous with 'upon which everything else is built'.

Nucleosynthesis is fundamental to atom-based life on Earth but I'm not convinced we could say it's fundamental to day-to-day life.

 

[Kajahtava]

I love it when people who just disagreed with me randomly re-stated what I said on another point in different words.

 

[humanino]

But what is proof theory then, or lambda calculus, or set theory, or recursion theory?

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".

 

[Kajahtava]

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".

Hmm, maybe this is your own environment, most I know are foundational mathematicians, but then again, that could be my own environment.

A forum like phyicsforums of course naturally draws nonfoundational maths. But is it truly that obscure or esoteric? Or well, you might have a point, seeing that when I still studied, one of the reasons I dropped out was that I found that a variety of courses I held to be foundational and essential were either optional or not given at all.

 

[humanino]

Maybe an objective answer could be obtained by doing some statistics on the arXiv preprints (for instance).

 

[DaveC426913]

I love it when people who just disagreed with me randomly re-stated what I said on another point in different words.

I assume you mean me.
1] We simul-posted. You beat me by a couple of minutes. It happens on popular boards, and around here quite a bit. I've only just seen your post now.
2] Just because I disagree with a point you've made doesn't mean I automatically agree with your opponent. I thought Jack's point about the definition of fundamental was a weak rebuttal, for the reasons I stated.

 

[Jack21222]

I assume you mean me.
1] We simul-posted. You beat me by a few seconds.
2] Just becasue I disagree with a point you've made doesn't mean I automatically agree with your opponent. I thought Jim's point about the definition of fundamental was a weak rebuttal for the reasons stated.

1) My name's not Jim.

2) My rebuttal was merely pointing out there is a difference in semantics in this argument.

Kaj is arguing that calculations and approximations aren't math. The rest of the world would claim that it is. Also, for many people, calculus is the basis for the "math" that they use every day. Kaj would call it "physics" or "biology," instead of math. Most real-world applications for math I can think of have their foundation in calculus.

Physics, engineering, economics, biology, statistics... none of these would be in its current state without calculus, and most people consider calculus as a type of math.

Therefore, I'd argue calculus is a "fundamental" type of math, even if it isn't fundamental "to" math.

 

[DaveC426913]

1) My name's not Jim.

What are you talking about. Are you hallucinating?

2) My rebuttal was merely pointing out there is a difference in semantics in this argument.
...
Therefore, I'd argue calculus is a "fundamental" type of math, even if it isn't fundamental "to" math.

You can argue that now, but when you stated it, you misstepped, asking about "fundamental to day-to-day life", which is a wild goose.

 

[Jack21222]

What are you talking about. Are you hallucinating?

You can argue that now, but when you stated it, you misstepped, asking about "fundamental to day-to-day life", which is a wild goose.

Agreed, I didn't make my intended point.

 

[Kajahtava]

I assume you mean me.
1] We simul-posted. You beat me by a couple of minutes. It happens on popular boards, and around here quite a bit. I've only just seen your post now.
2] Just because I disagree with a point you've made doesn't mean I automatically agree with your opponent. I thought Jack's point about the definition of fundamental was a weak rebuttal, for the reasons I stated.

Oh, I wasn't sarcastic, I love it when people see point 2.

The enemy of the enemy is only a friend in a dilemma. If only one of two options can be true, and if one is false, it implies the other is true, many people feel to appreciate this and I like it that you did.

Also, I think there is a difference between coming to a conclusion that is true, but not in a rigorous way and coming to a conclusion that it false by lack of rigour. As in, many results obtained by use of the infinitesimal are true. However the theory of the infinitesimal itself is an awkward thing.

 

[dx]

Einstein did not discover any new mathematics. Newton was one of the greatest mathematicians of all time, and certainly the foremost mathematician (and physicist) in the world during his lifetime.

 

[Kajahtava]

Didn't Einstein to some degree helped with tensor analysis?

Not sure about that one though.

 

[dx]

Well, he was the main reason tensor analysis entered the world of physics, but he didn't personally contribute anything to the mathematics. It was discovered and developed in the 1800s by Gauss, Riemann etc.

 

[Kajahtava]

Ahh, never knew it was that old.

Which brings us to another point though, if you ask a random person who the greatest mathematicians were, the response will almost always be 'Newton, Euler, Gauss, Riemann', that group of people. In fact, rarely will the response include a person that contributed profoundly to pure maths.

I think people have a tendency to greatly confuse brilliance / sophistication of work and importance of it.

 

[dx]

Ahh, never knew it was that old.

Which brings us to another point though, if you ask a random person who the greatest mathematicians were, the response will almost always be 'Newton, Euler, Gauss, Riemann', that group of people. In fact, rarely will the response include a person that contributed profoundly to pure maths.

I think people have a tendency to greatly confuse brilliance / sophistication of work and importance of it.

...What? Are you saying Euler, Gauss and Riemann did not contribute profoundly to pure math? I'm sorry but that's the most rediculous statement I've ever heard.

 

[Kajahtava]

That is exactly what I'm saying, they all contributed solely to applied mathematics. In fact, pure mathematics hardly existed in that era, it was about application, application and application and rigour was second to that.

Pure mathematics mainly got off the ground with people like Hilbert, Dedekind, Gödel, Frege, Russell, Cantor, for some reason a lot of German people...

 

[dx]

'Rigor' and 'pure math' are not synonyms. Number theory only became an applied field in the late 20th century, but Gauss, Euler and Riemann contributed greatly to it. Differential geometry only became an applied field after Einstein, but Gauss, Euler and Riemann contributed fundamentally to it.

 

[Kajahtava]

Frankly, I'd call any mathematics that deals with quantities and magnitudes and how to calculate them with some tricks 'applied'. It has an application outside serving as a lemma then, doesn't it?

 

[MotoH]

I voted for Alby because I figured he wouldn't have a lot of votes.

 

[M Grandin]

Maybe somewhat off-topic, but regarding thread title "Competition between geniuses"
and Newton involved, there is a very interesting youtube video about Robert Hooke in 6 parts, where first part at address

http://www.youtube.com/watch?v=KP7zeoYv1ZI&NR=1