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Lessons I wish I had learned before teaching differential equations [pdf] (1997)

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Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#201

Post by jampekka »

nuancebydefault wrote: post #191
DoughnutHole wrote: post #26

Earlier quoted context omitted.

He’s right - intuitive physical reasoning is useful (and often valid in the applied sciences) but it isn’t rigorous mathematical proof. Some areas of mathematics are applicable to physical realities, but they aren’t defined by those realities.

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?

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_Effectivene...

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#202

Post by anon291 »

gbjw wrote: post #195
anon291 wrote: post #175

Earlier quoted context omitted.

It really depends what you consider 'real'. It is actually pretty straightforward to realize a representation of 10^241 (you just did). Why does that make it any less real than the digit 2, which represents two things, but is actually just itself one thing. > Another related notion is that there is no way to 'realize' an irrational 'number' -- sqrt(2) can be defined but we can never draw a length of sqrt(2). Again, i…

> It really depends what you consider 'real'. It is actually pretty straightforward to realize a representation of 10^241 (you just did). Why does that make it any less real than the digit 2, which represents two things, but is actually just itself one thing. One is a map, and the other is the territory. Both 'real' in some sense but a map without the territory feels less 'grounded' (pun?). > Again, it's unclear what…

> Irrational numbers can be defined but not 'realized' (or 'constructed') in the same way that the rationals can.

Once again, I'm not 100% sure what you're saying. If you have something that's of length 1, then you can easily construct the line with a ratio sqrt(2):1. Draw another line of length 1 (use compass and straightedge) at a 90 degree angle. Repeat 4 times until you have a square. Now draw the diagonal. You have successfully constructed the square root of 2 in your own setup.

There are numbers that cannot be constructed geometrically using only a compass and straightedge (e for example). However, again these are no less 'real'. Drawing lines is not the only measure of real. There are other methods. In computing, we often say a number is computable if you can define a function that, given any rational number can tell you if the number it represents is greater than or equal to the rational number. The square root of two and e and pi, etc are easily representable this way. There are some numbers that are not, and perhaps these can truly be said to not exist.

However, the field of computables is closed anyway, so it really doesn't matter if you don't want to believe the reals exist.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#203

Post by anon291 »

ajkjk wrote: post #178
anon291 wrote: post #170

Earlier quoted context omitted.

> I cannot overstate how destructive this is, because it basically presents math as this arbitrary logic game rather than as a fundamental language of the universe. You are making a clear philosophical assumption yet don't realize it. Is math really the 'fundamental language of the universe'? or is it an arbitrary logic game that, in its most common interpretation, describes the universe well? Given that we have no f…

Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs. A simple example is: nobody needs a proof of how differentiation or integration work. They need to know how to do it, think with it, apply it to problems, a…

> Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs.

That's great. Not everyone does. Thinking simply via formal systems is not inherently inferior to 'doing' something with it. Both have their uses. In particular, as I said, quantum mechanics would not be where it's at today if it weren't for many pre-quantum mathematicians pushing around symbols on a page.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#204

Post by Koshkin »

Workaccount2 wrote: post #101
ajkjk wrote: post #16

Earlier quoted context omitted.

Physics doing the work of actually teaching math is a universal embarrassment to math departments. Or, well, it should be.

Many math teahcers (and likely mathematicians) could not care less about the real world applications of math. My college calculus teacher had no interest in physics or science. She liked math in the same way someone would like doing a crossword or Sudoku. A puzzle game with a well defined rule set and the challenge of finding answers. I cannot overstate how destructive this is, because it basically presents math as t…

I would not call this "destructive." Solving problems is fun in its own right, and things would start making sense in their interconnection, regardless of whether they can be applied to sciences or not.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#205

Post by sampo »

nuancebydefault wrote: post #191
DoughnutHole wrote: post #26

Earlier quoted context omitted.

He’s right - intuitive physical reasoning is useful (and often valid in the applied sciences) but it isn’t rigorous mathematical proof. Some areas of mathematics are applicable to physical realities, but they aren’t defined by those realities.

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?

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

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#206

Post by guhidalg »

nuancebydefault wrote: post #191
DoughnutHole wrote: post #26

Earlier quoted context omitted.

He’s right - intuitive physical reasoning is useful (and often valid in the applied sciences) but it isn’t rigorous mathematical proof. Some areas of mathematics are applicable to physical realities, but they aren’t defined by those realities.

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?

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

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#207

Post by Koshkin »

ajkjk wrote: post #178
anon291 wrote: post #170

Earlier quoted context omitted.

> I cannot overstate how destructive this is, because it basically presents math as this arbitrary logic game rather than as a fundamental language of the universe. You are making a clear philosophical assumption yet don't realize it. Is math really the 'fundamental language of the universe'? or is it an arbitrary logic game that, in its most common interpretation, describes the universe well? Given that we have no f…

Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs. A simple example is: nobody needs a proof of how differentiation or integration work. They need to know how to do it, think with it, apply it to problems, a…

> nobody needs a proof of how differentiation or integration work

This "nobody" includes a lot of industry workers, e.g. those in finance and economics, not only "pure" mathematicians.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#208

Post by nuancebydefault »

ink_13 wrote: post #199
nuancebydefault wrote: post #193

Earlier quoted context omitted.

> Study of things that are provably true. Why would you study such trueness if whether you know it is true or not does not have any added value in the physical world?

I don't even understand the question. Is there in truth no beauty? Why do anything that doesn't add value to the physical world?

It seems you did understand the question and I thank you for your answer, it is helpful.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#209

Post by nuancebydefault »

jampekka wrote: post #201
nuancebydefault wrote: post #191

Earlier quoted context omitted.

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?

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…

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

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#210

Post by nuancebydefault »

sampo wrote: post #205
nuancebydefault wrote: post #191

Earlier quoted context omitted.

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?

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

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.
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