Why Is Math So Useful?

15 October 2021

Is math a realm of timeless, universal truths? Or are mathematicians just making it up as they go? If equations are made up, why are they so useful? We’ll be discussing these questions, and more on this week’s episode, “The Mysterious Timelessness of Math.”

Math is obviously good for many things: we use it for everything from building bridges to designing lasers to predicting the motions of planets to explaining why snowflakes have that odd six-pointed shape. But why is it good for so many things?

Maybe it describes the fundamental structure of the universe. But that makes its methodology look puzzling, on the face of it: how can anyone learn about the fundamental structure of the universe just by scribbling symbols on a whiteboard or a piece of paper? Mathematicians don’t run experiments. They don’t even write down observations about the physical world.

Some would say that there’s no real puzzle here: if math is latching on to deep truths (in particular, the kind that philosophers call necessary truths) then maybe we don’t need evidence to trust it. A claim like “I’m wearing purple socks” isn’t a necessary truth. It’s contingent: even if it happens to be true, it could easily have been false. In order to know whether it’s true, you have to actually check. But for a necessary truth, like 2+2=4, there’s no need to check, because no circumstance could possibly make it false.

But this reply isn’t entirely adequate. We do have ways of checking whether something is a truth of mathematics; mathematicians develop proofs, and try to generate counterexamples to important conjectures. It’s just that those methods typically don’t involve any reliance on physical experiments (although the rise of computers has changed this, making some branches of mathematics increasingly reliant on Monte Carlo methods and computer-assisted proofs). 

Another possible answer to the question “how do we know that math describes the fundamental structure of the universe?” is that math works.  Math is indispensable for the electronics that brought this blog post to you: if you want to build a computer, you need to understand electromagnetism, and you can’t do that without equations. But this answer doesn’t seem totally adequate either: the indispensability of math might be a good reason to believe that it’s latching on to important truths, but it still doesn’t explain how math is capable of latching onto such important truths.

Another response to the challenge is to say that math doesn’t give us factual information at all. Instead, it’s a useful filing system for organizing what you know—one where you still have to add the information yourself. This raises some worries about arbitrariness (why not pick an organizing system where 2+2= 5, if that’s convenient?), but we can cut down on some arbitrariness by requiring that mathematical systems be internally consistent. Regular arithmetic, where 2+2=4, is not inherently better than mod-3 arithmetic, where 2+2=5; they're both internally consistent systems but with different uses.

I think that reconciling the usefulness of math with its methods is going to take some more philosophical work! I’m looking forward to exploring possible answers with our guest this week, philosopher Arezoo Islami from San Francisco State University.

Photo by Jeswin Thomas on Unsplash

Comments (6)


Tim Smith's picture

Tim Smith

Friday, October 15, 2021 -- 5:14 PM

There are very profound ways

There are very profound ways in which “2+2=5”, but modulo 3 is not one of them. There “2+2=1” or “2+2=-5”. All three of these statements have mathematical truth, as does “2+2=4”, and even more literary and philosophical depth.

There is a perceptual truth to the color of socks that begs the contingent fact as well.

Math as a metaphor for nature is deceptively accurate. Einstein and Tagore had a bout about this. Eugene Wigner was troubled by it. Arezoo had much more to say and misspoke a bit or would say more herself. Math is a hammer.

I liked this show and the topic, but as usual, there wasn’t enough time. That Islami thinks about time is prophetic of perhaps a future blog or show where she could shed light on time and math grazed and left wanting here.

Thanks, Ray, for the recommendation on Yackel’s book. I will attempt to figure that out over winter break.

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MJA's picture

MJA

Sunday, October 17, 2021 -- 10:34 PM

2 + 2 = X, 3+1, #, or

2 + 2 = X, 3+1, #, or anything else, even a blank space. But there is one equation that is absolute, that unites everything, making everything just One. It is the equation that Einstein died searching for. And it was simply time that stood in his way, imagine that. = is the solution to his quest. = is the light at the end of the tunnel, the promised land, the truth.

What do we fight for? What did MLK, Gandhi, and Lincoln die for? Equal is the definition of justice. And what about our Democracy, it is based on equality. Yet for some reason, whilst we focus mathematically on the left and right side of an equation, = the truth or absolute is over-looked.

Has anyone had a class called Equal 101, anyone? We are taught mathematics but not equality. What is up with that? Maybe that is why we struggle with it so.

To answer the above question: equal is the Universal Truth.
Thanks, =

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Tim Smith's picture

Tim Smith

Thursday, October 21, 2021 -- 12:57 PM

That is what Einstein's first

That is what Einstein's first wife said!

I kid, somewhat, but Einstein struggled with family/social norms, social justice, anti-semitism, and equality his entire life. Equality there did not match the equality of the unified theory that he pursued late in life. The devil is in the details.

Einstein was a complicated man and cultural figure. He lost his first child to potential parental neglect without ever meeting her (it seems) despite being a long but doable train ride away. She died at 21 months, the details of which are not clear. Walter Isaacson's biography is fairer to this than I am here.

It is funny that mathematics and Einstein run together so often, probably due to the visual simplicity of e=mc^2. Alfred was no natural mathematician (very few people are), and his first wife did help him with it. Einstein excelled at thought experiments that were not based on math for much of his lauded work.

Equality in math is vital, as is symmetry. However, that supersymmetry was not demonstrated at the large hadron collider points back to the still excellent standard model. I doubt time held back Einstein or the world. We have sunk many billions of dollars and brain hours looking for that equality sign. This may be due to a philosophical bias toward an a priori mathematical reality. That this reality is not necessarily present is reflected in some of Arezoo Islami's work. Not an "=", but a "?".

Math has never been as strong as it is today. Linear Algebra is running computational roughshod over previously intractable practical problems. Absolute top intellect is engaged and publishing preprints and communicating in a collegial environment that very much aims toward equality. Shinichi Mochizuki's proof of Szpiro's conjecture was published just this year in a mind-bending fashion that will take experts years to validate. We all should be celebrating co-existence with Terence Tao, Ed Frenkel, and Kiran Kedlaya et al. My personal favorite is Steven Strogatz. He has written many books for non-math and math types to explain the wonder and practicalities.

Math research is cheap. When we support it properly, it can be fantastically rewarding. When we philosophically call it a prescriptive reality, that calls for a rebalancing of the scales. These are human beings using human tools to look at and describe their own thoughts and nature.

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MJA's picture

MJA

Friday, October 22, 2021 -- 11:19 PM

I would prefer to talk to The

I would prefer to talk to The Professor, but alas he is gone and you will have to do. There is some jest here too.
I feel better now.
As for time keeping Einstein from the truth he died searching for, it is true. He said it was Maxwell that taught him the universal constant of the speed of light. A constant that is probable at best. Had he understood that nature including light is truly immeasurable, as science itself has proven, QM or probability at best, he would have removed it from his famous equation. God does not play dice! Oh, now we are back to Copenhagen, from whence science has never left, nor resolved. For shame.
The flaw in science is in its very foundation, measure. "Man is the measure of all things," Protagoras said. The flaw in the scientific method of measure is spelled out in Heisenberg's Uncertainty Principle. Ye must have faith in science too!

What I am getting at here is the resolution of the QM problem, the solution to the Unified Field Theory, the answer to Einstein's quest, And, the proof of God! Truth is

If energy equal mass times the speed of light squared, but light is truly immeasurable, then his equation can be reduced by removing the uncertainty, leaving energy equals mass. And if energy equals mass then absolutely = is all that remains. In fact it is the only certainty in an equation. It is also the ultimate solution.
The mathematical equation for everything is =. = is the single truth.

As Einstein knew, simplification was the path to the solution. It was "C" that stood in his way.

How important is it you ask? it is what mankind has been fighting and dying for. MLK, Gandhi, Lincoln, Justice, Democracy, Civil War, BLM, uh everything. And science can't find the equation that unites everything, WOW!

Oh and I am so glad you mentioned the collider, my favorite. The solution is not dividing minute particles to find the God particle, the solution was in the unity of all particles, including space, science went the wrong way. Oneness anyone. What is more powerful than everything, another name for God?

Can One teach the blind to see, or must One learn to see Oneself? I think the answer here is the later, but then One can only TRY!

Be One, =

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Tim Smith's picture

Tim Smith

Saturday, October 23, 2021 -- 7:54 AM

MJA,

MJA,

There is no replacement for The Professor. I will not due, or do, for that matter. If that makes you feel better, you are a better person than I am. Knowing myself, that is a shallow bar. I make mistakes, change my philosophy, and rewrite my life with every word and learning.

As you say, one must learn to see oneself. It is good to understand the multiple meaning and import in that statement.

Math is a human device, as are algorithms. When we look with math to see, is it any wonder that we end up in Nick Bostrom’s simulation hypothesis. How do we refer to the world? That is a question left to us by others.

Though symmetry is fundamental to math, equality is not nor even to human culture. Equality is a tool that allows insight. The majority of humans show positive bias/affect/valence for those who look like them, not their equals. As you say, it is a struggle and a worthy one.

The majority of scientists, technologists, and capitalists, who drive our world, philosophically misunderstand mathematics and algorithms as separate from human tools. Elon Musk is one. Getting philosophical issues wrong is never good, always wasteful, and rarely serves the greater good.

I am confident we both aspire to the greater good, though I am not your equal.

Thanks for this get back. I respect your view and insight and give it greater weight than my own. It is a better world than I see. It is one I seek as well.

Tim

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MJA's picture

MJA

Saturday, October 23, 2021 -- 8:57 AM

Equal is =, as one is One, be

Equal is =, as you is I, as 1 is One.
I can't thank you enough,
Be One. =

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