Why Does Mathematics Work?

Spearthrower · #1 by **Spearthrower** » May 04, 2019 8:01 pm

Great answer, especially as he acknowledges he doesn't really know either! :cheers:

Cito di Pense · #2 by **Cito di Pense** » May 04, 2019 8:18 pm

These questions are fun, and Wigner wrote a very fine essay. If we strip away all the filosofeezing, we are still only asking why something works. Maybe a better question is "why does mathematics work so well?" People try to make things work, and there are lots of innumerate people who still try to make things work (that haven't been designed for them). We might as well ask why we want stuff that works, as opposed to stuff that doesn't work. I mean, we do stuff for fun, but somehow, fun works.

At the simplest level, mathematics is a project for making knowledge systematic, where we try as hard as we can to ensure that what we come up with will work. Why does it work? Why ask why?

Wigner asked why it works so well in the physical sciences, instead of just as a project to try as hard as we can to make something work for its own sake, and that is a deep mystery. We live in this universe and want to know how it works, and we found a tool that is effective in getting the know-how together. There's always that picture that Penrose drew in the first few pages of The Road To Reality.

I confess, I didn't watch the video all the way through, after observing the limited effectiveness of the interviewer, and having first read Wigner's essay many years ago. As Ellie Arroway said, "they should have sent a poet".

Spearthrower · #3 by **Spearthrower** » May 04, 2019 8:53 pm

Cito di Pense wrote:Why ask why?

I think it ends up being the most intriguing of questions. All the other wh- questions have such immediate, definitive and mundane answers, but whys can take you on long journeys into realms you hadn't imagined when you originally asked. Kids seem to alight on the why question at a particular age and recognize its power, or perhaps its futility - no other question requires quite so much effort or leaves the adult flapping so much.

Cito di Pense wrote:There's always that picture that Penrose drew in the first few pages of The Road To Reality.

Do you mean the 3 intersecting spliffs of reality?

Cito di Pense · #4 by **Cito di Pense** » May 04, 2019 9:05 pm

Spearthrower wrote:
Do you mean the 3 intersecting spliffs of reality?

Yeah, that. I decided that business is all smoke and, uh, mirrors. You know, "lines on the mirror, lines on her face..."

tuco · #5 by **tuco** » May 04, 2019 9:24 pm

I guess it's a similar question to why magnets work?

Spearthrower · #6 by **Spearthrower** » May 04, 2019 9:30 pm

Not really, no.

Feynman's response to the question is to explain that the interviewer expects a proximate answer but to actually answer the question requires giving ultimate answers which will completely and perfectly explain why magnets work, but will simultaneously not answer the question at all given the interviewer wants a proximate answer.

Why magnets work is purely physical. Why mathematics works is not so straight forward.

tuco · #7 by **tuco** » May 04, 2019 9:42 pm

The only thing that can be explained is how something works, not why.

aufbahrung · #8 by **aufbahrung** » May 04, 2019 11:01 pm

Doesn't work except within a narrowly defined set of parameters. Most equations are duds. And even those with the 'right' answer could still cause a plane crash....in the wrong place.

laklak · #9 by **laklak** » May 04, 2019 11:10 pm

Because God, of course. Remember, pi was 3 back in biblical days. That's because we didn't have calculators back then, or a lot of pencils and paper. Now that we do God changed it to it's current value. That's why we can't rebuild the ark without pre-stressed concrete and high tensile steel, the maffs have changed.

tuco · **#10** by **tuco** » May 04, 2019 11:57 pm

Indeed, because that is how the universe seems to be. Why does anything work, right? Now we can try to explain how anything works, going to the greatest detail, but sooner or later we hit a wall: irreducible fact. So now we ask why does this irreducible fact exist?

Spearthrower · **#11** by **Spearthrower** » May 05, 2019 5:46 am

tuco wrote:The only thing that can be explained is how something works, not why.

Linguistics isn't as important as some people think it is.

Spearthrower · **#12** by **Spearthrower** » May 05, 2019 5:48 am

aufbahrung wrote:Doesn't work except within a narrowly defined set of parameters. Most equations are duds. And even those with the 'right' answer could still cause a plane crash....in the wrong place.

Which basically agrees that there is something correct attainable by mathematics.

Of course, no one said that all mathematics is correct.

aufbahrung · **#13** by **aufbahrung** » May 05, 2019 7:13 am

Spearthrower wrote:
aufbahrung wrote:Doesn't work except within a narrowly defined set of parameters. Most equations are duds. And even those with the 'right' answer could still cause a plane crash....in the wrong place.

Which basically agrees that there is something correct attainable by mathematics.

Of course, no one said that all mathematics is correct.

Can we remove statistics from the realm of mathematics before leaving 'math is magic' for 'math is God' to begin with?

Cito di Pense · **#14** by **Cito di Pense** » May 05, 2019 7:31 am

aufbahrung wrote:
Can we remove statistics from the realm of mathematics before leaving 'math is magic' for 'math is God' to begin with?

What can you mean by this? Do you know anything about the mathematical underpinnings of probability distributions?

Spearthrower wrote:
aufbahrung wrote:Doesn't work except within a narrowly defined set of parameters. Most equations are duds. And even those with the 'right' answer could still cause a plane crash....in the wrong place.

Which basically agrees that there is something correct attainable by mathematics.

But what is that? There are equivalences between pairs or sets of abstract (mathematical) objects. "Correct" refers to a correspondence between a statement and 'reality' or else adherence to agreed-upon rules. After a certain point not long after you start, mathematics is constructed, although you can have arguments about that if filosofeezing is your thing.

We (at the RatSkep level) are either counting events or members of sets or else we are approximating real (or complex) numbers, which is all we can do by computing. Yes, we can get the right correspondences between two sets. The banker with the red car does not own a dog. What color hat is worn by the woman with the brindled guinea pig? If the problem is set up right, there is a correct answer, and we can select it on a multiple-choice exam.

Spearthrower · **#15** by **Spearthrower** » May 05, 2019 7:50 am

aufbahrung wrote:Can we remove statistics from the realm of mathematics before leaving 'math is magic' for 'math is God' to begin with?

I'm not sure I understand anything you wrote.

Spearthrower · **#16** by **Spearthrower** » May 05, 2019 8:08 am

Cito di Pense wrote:
But what is that? There are equivalences between pairs or sets of abstract (mathematical) objects. "Correct" refers to a correspondence between a statement and 'reality' or else adherence to agreed-upon rules. After a certain point not long after you start, mathematics is constructed, although you can have arguments about that if filosofeezing is your thing.

One of the things I find intriguing about mathematics, while simultaneously admitting not being able to understand it, is that while we can obviously describe things we observe with mathematics, we can also describe things that cannot be observed.

To use the old analogy of the map and the terrain but to add more fecal matter references to its composition; in most attempts at understanding shit, we first observe shit, then we make a mental model of shit, and how well the map corresponds to contemporary and future observations is its accuracy - this of course supposes a shit-in-itself independent of observations, and consequently acknowledges that our constructed maps may never attain 100% fidelity. The map is of something that is, and even if we haven't yet seen all there is of it, we can still make some neat and useful descriptions of it that may even let us predict how similar shit behaves under similar circumstances, or an expectation of shit we've yet to observe.

But mathematics goes a step further and generates from itself as-yet-unobservable-shit, even potentially never-observable-shit, and makes predictions from that which sometimes then even gives us higher fidelity in descriptions of the shit that we can observe. There must be a maximal fidelity model, and while we can never draw a perfect line, or scale things perfectly, at least notionally, we should or at least could be able to attain that mathematically.

I'm not the best person to be philosophizing or filosofeezing about mathematics. I believe I very quickly achieved the first forum award for Rampant Nematodery within the realm of mathematics.

http://www.rationalskepticism.org/post1 ... atode#p186

newolder · **#17** by **newolder** » May 05, 2019 8:15 am

If mathematics didn't work at all then my current Posts count would be 987654321 and after I click "Submit" it would change to eleventy spark and later become 6000 for no reason at all.

Mathematical statements that are true but unprovable from within the current axioms are inevitable Gödelian constructions. So what? A Circus sans Freak Show is hardly worth the visit.

Spearthrower · **#18** by **Spearthrower** » May 05, 2019 9:03 am

newolder wrote:If mathematics didn't work at all then my current Posts count would be 987654321 and after I click "Submit" it would change to eleventy spark and later become 6000 for no reason at all.

Sure - I think the weak-anthropic principle provides some insight as to why it works on a most abstract level: because the universe has to work in some fundamentally consistent way in order for things like us to exist and contrive methodologies to decipher it.

tuco · **#19** by **tuco** » May 05, 2019 9:05 am

This is the essay:

The Unreasonable Effectiveness of Mathematics in the Natural Sciences - http://www.dartmouth.edu/~matc/MathDram ... igner.html

It opens with:

THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."

and follows later with this:

Most of what will be said on these questions will not be new; it has probably occurred to most scientists in one form or another. My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. In order to establish the first point, that mathematics plays an unreasonably important role in physics, it will be useful to say a few words on the question, "What is mathematics?", then, "What is physics?", then, how mathematics enters physical theories, and last, why the success of mathematics in its role in physics appears so baffling. Much less will be said on the second point: the uniqueness of the theories of physics. A proper answer to this question would require elaborate experimental and theoretical work which has not been undertaken to date.

for those who are not willing to read further, Ctrl+F "why" = 2 hits:

.. why the success of mathematics in its role in physics appears so baffling.
.. we do not know why our theories work so well.

Now, can we imagine a universe where pi wasn't like the pi we know? Of course, we can. Would our math work in such a universe and why?

Does anyone know the story of integral? Why and how it was invented or discovered? Or Pythagoras theorem?

Thommo · **#20** by **Thommo** » May 05, 2019 10:31 am

tuco wrote:Now, can we imagine a universe where pi wasn't like the pi we know? Of course, we can.

Can we?

If we lived in a curved universe (or a curved part of our universe) then attempts to draw a circle and measure it would not result in empirical conclusion of the value of pi we recognise, but why would we say the value of pi changed? The ratio of a cartesian circle's radius to its circumference would not have changed, so if we consider that (or an equivalent power series based definition) to be pi, then we haven't imagined the universe we were after, after all.

Why Does Mathematics Work?

Why Does Mathematics Work?

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