4 ms·

If some mathematics are not applicable to physical realities, to what are they applicable? Is there any un-applicable math and if so, why does it even exist?

jampekka 3y ago
Maybe depends on what's meant by application, but I'd say a lot of, probably most, research-level math has no (clear) practical applications and most research is not done from the PoV of applications or relationship to physical reality. Math is study of formal systems, sort of "what can and can't follow from some set of formal rules". It's more a surprise that some rulesets map to physical world as well as they do [1]. For some there may be very important ones in the future like e.g. number theory got in cryptography. But for example just knowing whether P=?NP or if the Riemann hypothesis is true probably has no "physical reality" direct applications. [1] https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectivene...
- nuancebydefault 3y ago
  So if the application is not visible today, it might appear tomorrow. In fact that makes sense.
sampo 3y ago
Up until about World War II and use in cryptography, number theory was pretty much an un-applicable field of mathematics.
- nuancebydefault 3y ago
  Pretty amazing that someone discovered and explored number theory without knowing if there would be an application to it. Hopefully they lived to see its usage.
  - jampekka 3y ago
    Number theory was a big deal already in ancient greece and probably got there from Babylonia, so it's probably safe to say that they didn't live to see the public key crypto usage. :) But (integer) numbers and their behavior had huge significance in the ancient worldview, and still do even in our days if you look deep enough. And the Pythagoreans et al applied number theory in e.g. music.
- The_suffocated 3y ago
  I may be wrong but this seems inaccurate. In WWII, the cryptanalysis of the Enigma (https://en.wikipedia.org/wiki/Cryptanalysis_of_the_Enigma https://en.wikipedia.org/wiki/Cryptanalysis_of_the_Enigma) was related not to number theory, but to group theory. Number theory became relevant to cryptography only since the 1970’s/1980’s when Diffie and Hellman published their paper on public-key cryptography. Although number theory was related to cryptography in the past (e.g. in shift ciphers and block cyphers), it was the Diffie-Hellman paper that placed number theory in the central role.
guhidalg 3y ago
Fiction writing exists, should it not be enjoyed? Sometimes mathematicians invent things before scientists discover how to apply it.
DoughnutHole 3y ago
The biggest example I can think of is Fermat's Last Theorem. Uncounted mathematicians devoted years of their lives to the theorem, and the developments they made along the way may have some applications. But I have yet to hear of any physical truths derived from the fact that a^n + b^n != c^n for n > 2. Most math is like this - beautiful, and useless. The motivation of most non-applied mathematicians is aesthetic, much like the motivation of someone in the arts. It's done for its own sake ("mathematical beauty"), not because it has any practical application. Mathematics might turn out to be "useful" more often than the average novel - but that doesn't mean the math was done with any use in mind. For example many developments in differential geometry made in the 19th century were inapplicable to anything physical for decades until Einstein applied them to create general relativity. Plenty of math becomes useful long after its development, and plenty more never finds any practical application.

jampekka 3y ago

Maybe depends on what's meant by application, but I'd say a lot of, probably most, research-level math has no (clear) practical applications and most research is not done from the PoV of applications or relationship to physical reality. Math is study of formal systems, sort of "what can and can't follow from some set of formal rules". It's more a surprise that some rulesets map to physical world as well as they do [1]. For some there may be very important ones in the future like e.g. number theory got in cryptography. But for example just knowing whether P=?NP or if the Riemann hypothesis is true probably has no "physical reality" direct applications. [1] https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectivene...

nuancebydefault 3y ago
So if the application is not visible today, it might appear tomorrow. In fact that makes sense.

sampo 3y ago

Up until about World War II and use in cryptography, number theory was pretty much an un-applicable field of mathematics.

nuancebydefault 3y ago
Pretty amazing that someone discovered and explored number theory without knowing if there would be an application to it. Hopefully they lived to see its usage.
- jampekka 3y ago
  Number theory was a big deal already in ancient greece and probably got there from Babylonia, so it's probably safe to say that they didn't live to see the public key crypto usage. :) But (integer) numbers and their behavior had huge significance in the ancient worldview, and still do even in our days if you look deep enough. And the Pythagoreans et al applied number theory in e.g. music.
The_suffocated 3y ago
I may be wrong but this seems inaccurate. In WWII, the cryptanalysis of the Enigma (https://en.wikipedia.org/wiki/Cryptanalysis_of_the_Enigma https://en.wikipedia.org/wiki/Cryptanalysis_of_the_Enigma) was related not to number theory, but to group theory. Number theory became relevant to cryptography only since the 1970’s/1980’s when Diffie and Hellman published their paper on public-key cryptography. Although number theory was related to cryptography in the past (e.g. in shift ciphers and block cyphers), it was the Diffie-Hellman paper that placed number theory in the central role.

guhidalg 3y ago

Fiction writing exists, should it not be enjoyed? Sometimes mathematicians invent things before scientists discover how to apply it.

DoughnutHole 3y ago

The biggest example I can think of is Fermat's Last Theorem. Uncounted mathematicians devoted years of their lives to the theorem, and the developments they made along the way may have some applications. But I have yet to hear of any physical truths derived from the fact that a^n + b^n != c^n for n > 2. Most math is like this - beautiful, and useless. The motivation of most non-applied mathematicians is aesthetic, much like the motivation of someone in the arts. It's done for its own sake ("mathematical beauty"), not because it has any practical application. Mathematics might turn out to be "useful" more often than the average novel - but that doesn't mean the math was done with any use in mind. For example many developments in differential geometry made in the 19th century were inapplicable to anything physical for decades until Einstein applied them to create general relativity. Plenty of math becomes useful long after its development, and plenty more never finds any practical application.