I want to ask some pure mathematician if the study of mathematics without the deep confrontation with the physical reality is not in some way frustrating. I cannot imagine learning some ideas and making proof of them without the immediate application of them. Do you think this confrontation could in any way help to understand the mathematical concept or do you think that physical reality is not needed for better understanding.
Pure mathematics...
| redace wrote: |
| I want to ask some pure mathematician if the study of mathematics without the deep confrontation with the physical reality is not in some way frustrating. I cannot imagine learning some ideas and making proof of them without the immediate application of them. Do you think this confrontation could in any way help to understand the mathematical concept or do you think that physical reality is not needed for better understanding. |
Speaking as a non expert I can imagine one possible reply:
it is actually liberating rather than frustrating. You are free to explore ideas and patterns and relationships without the need to tie these to messy physical relationships and quantities, therefore the real beauty of the expressions is revealed.
This says very little about a 'mathematical concept' since math is essentially a language and science uses the language in whatever field is in question to describe its own system of laws and relationships. Math is used because it is unambiguous and logical.
Chris
I'm no expert either, but from the books I've read about famous mathematicians it seems like "pure" mathematics is much easier to prove and handle. Once a mathematical theory has been proven, it will stay proven forever.
It's the "real world" theories that's hard (even impossible) to prove. It suffices with one observation of a counter example to a "proven" theory and it has to be revised or rejected.
A great read if you want to get some insights to the mind of a mathematician : http://www.simonsingh.net/Fermat_Corner.html (no math skills required)
It's the "real world" theories that's hard (even impossible) to prove. It suffices with one observation of a counter example to a "proven" theory and it has to be revised or rejected.
| Quote: |
| An astronomer, a physicist, and a mathmetician are riding on a train in Ireland. They pass by a farm that has a lone white cow. The astronomer says "How about that, all cows in Ireland are all white." The physicist corrects him and says that some cows in Ireland are all white. The mathmetician corrects both of them by saying that in Ireland, there exists at least one cow that is at least half white. |
A great read if you want to get some insights to the mind of a mathematician : http://www.simonsingh.net/Fermat_Corner.html (no math skills required)
Those 2 guys are right. In an uninhibited realm of nothingness, mathematics is absolute and perfect. In the real world, many factors evolve into quite the complications.
Just like with science, you ideally want to have control over EVERY single spec before you test something.
Just like with science, you ideally want to have control over EVERY single spec before you test something.
| FunFunkyFritz wrote: |
| I'm no expert either, but from the books I've read about famous mathematicians it seems like "pure" mathematics is much easier to prove and handle. Once a mathematical theory has been proven, it will stay proven forever. |
No. Mathematics is a science, just like physics. Theories can never be proven true in any science, even math.
Mathematics is unique from natural sciences in that it is entirely theoretical, and based not on observations, but on axioms. All mathematics is is essentially just a base set of axioms that have been combined and built on and point to difficiences in the axioms which require further axioms. This is as opposed to a natural science, which stems from observations, which get combined and built on, and point to deficiencies which require further observations.
Because mathematics is axiomatically defined, it is possible to take some of the base axioms and derive from them new ideas, which, because they are derived directly from the basic axiomatic building blocks of mathematics, must be absolutely, perfectly true.
These axiomatically defined concepts are called theorems. There are lots of theorems in math, like the Pythagoream theorem, Taylor's theorem or Lagrange's mean value theorem.
Natural sciences are not axiomatically defined, so they do not have theorems. They do, however, have laws, which are not absolutely proven, but have been shown to be correct in all tested cases. In the natural sciences there are lots of laws, such as Kepler's laws, Gauss' law and so on.
Both math and science do have theories, which are collections of hypotheses used to describe or explain or define a concept. There are lots of theories in math, like set theory, chaos theory and game theory, and lots in the natural sciences, like evolution and the standard model of physics. Theories are not, and cannot be, proven.
A theory can be proven wrong, but a theorem cannot, because it is axiomatically defined. It's correct by definition. It's like defining a straight line as the shortest distance between two distinct points, then proving that a straight line is the shortest distance between two points. There's nothing to prove, it's right by definition. All a theorem is, really, is an aggregation of axioms. If the shortest distance betweem two points is a straight line, then the shortest distance between several points will be a set of straight lines drawn from one point to the next.
In mathematics, a proof is simply taking a relationship, and showing that that relationship is correct using axioms.
| FunFunkyFritz wrote: | ||
It's the "real world" theories that's hard (even impossible) to prove. It suffices with one observation of a counter example to a "proven" theory and it has to be revised or rejected.
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That's not a very good example. In all three disciplines, there is only one correct answer - assume all cows everywhere are all white, and point out that experimental evidence (actual experiment in the case of natural science, proven cases in the case of math) has confirmed that there is at least one half-white cow and there is no contradictory evidence.
I've always been fond of this one:
| Quote: |
| A biologist, an engineer, a physicist, and a mathematician were sitting outside of a building where they saw two people walk inside. An hour later, three people come back out.
The biologist asserts, "Reproduction!" The engineer shakes his head, "No, there is an infinite amount of people inside the building, and every so often one slips out." The physicist waves this off and says, "No, the discrepancy can be negleged because it was within experimental error tolerances." And the mathematician says, thoughtfully, "If exactly one person goes inside the building now, there will be zero people in the building." |
All of them are wrong, but each is wrong in their own, unique way.
Whilst not disagreeing with much, I do disagree about math being science.
There are certainly 2 schools of thought and both cases can be made I think.
Here is my favourite story on the matter:
(Stefan Bilaniuk,Department of Mathematics,Trent University, Ontario)
There are certainly 2 schools of thought and both cases can be made I think.
Here is my favourite story on the matter:
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Some academics relaxing in a common room are asked whether all odd numbers greater than one are prime. The physicist proceeds to experiment -- 3 is prime, 5 is prime, 7 is prime, 9 doesn't seem to be prime, but that might be an experimental error, 11 is prime, 13 is prime -- and concludes that the experimental evidence tends to support the hypothesis that all odd numbers are prime. The engineer, not to be outdone by a physicist, also proceeds by experiment -- 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, 13 is prime, 15 is prime -- and concludes that all odd numbers must be prime. The statistician checks a randomly chosen sample of odd numbers -- 17 is prime, 29 is prime, 41 is prime, 101 is prime, 269 is prime -- and concludes that it is probably true that all odd numbers are prime. The physicist observes that other experiments have confirmed his conclusion, but the mathematician sneers at "mere examples" and posts the following: 3 is prime. By an easy argument which is left to the reader, it follows that all odd numbers greater than one are prime. |
(Stefan Bilaniuk,Department of Mathematics,Trent University, Ontario)
there are inspiration mathematics, applied mathematics and pure mathematics.
as an ex disgruntled physics major and current computer scientist, i agreed with Nikolai Lobachevsky
if only we stick with inspiration mathematics (that is, you got problem now, you solve it now), frihost wouldn't exist by now.
as an ex disgruntled physics major and current computer scientist, i agreed with Nikolai Lobachevsky
| Quote: |
| there is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world. |
if only we stick with inspiration mathematics (that is, you got problem now, you solve it now), frihost wouldn't exist by now.
Lazy Undergrad * (Math + a computer) = Science
Sometimes, pure mathmatics comes about because of applied math. Newton wanted to know how gravity worked, so he created calculus. He didn't quite know how infinitesimals worked, but he made progress for more modern mathematicians in the abstract field of analysis.
On the other hand, the reverse sometimes happens as well. Moduli, primes, and integer exponentation seem pretty abstract and practially useless, but tell that to everyone who's ever made a purchase online. You need it for public-key encryption.
Just some thoughts to think about.
Sometimes, pure mathmatics comes about because of applied math. Newton wanted to know how gravity worked, so he created calculus. He didn't quite know how infinitesimals worked, but he made progress for more modern mathematicians in the abstract field of analysis.
On the other hand, the reverse sometimes happens as well. Moduli, primes, and integer exponentation seem pretty abstract and practially useless, but tell that to everyone who's ever made a purchase online. You need it for public-key encryption.
Just some thoughts to think about.
From my experience (somewhat limited), whether or not pure maths is considered boring or not depends largely on the person themselves and the aspect of maths that they are studying.
I must admit that there are times when you are undertaking calculations that seem pointless, however often the sheer ambiguity of the work creates intregue. THe most interesting aspect of pure maths is most certainly its ability to deal with calculations that have little specific purpose in the sciences, and which are more aimed at a general desire to work logically.
Despite this however, yes, it does get frustrating, occasionally!
I must admit that there are times when you are undertaking calculations that seem pointless, however often the sheer ambiguity of the work creates intregue. THe most interesting aspect of pure maths is most certainly its ability to deal with calculations that have little specific purpose in the sciences, and which are more aimed at a general desire to work logically.
Despite this however, yes, it does get frustrating, occasionally!
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