We’re always lumping them together, scientists and mathematicians. They’re “STEM” professionals: bespectacled, smart, pleasantly soft-spoken until you conflate *Star Trek* and *Star Wars*, after which their wrath is visited upon you.

But the fact is that, aside from being the butt of cheap jokes, mathematicians and scientists don’t share all that much in common.

And you can tell that from the way they look at each other’s fields.

When it comes to research, **scientists view mathematics the way a handyman views a toolkit**. To a scientist, math is a way of solving problems, as practical as a step-ladder or a roll of duct tape.

Want to describe an object falling to earth? Draw up a quadratic equation!

Want to investigate a rate of change? Take a derivative!

Want to model an electromagnetic attraction? Bring out the vector fields!

When people extol the “real-world” benefits of mathematics, they’re talking about moments like this, when a scientist employs quantitative techniques to analyze the world around us.

Meanwhile, **mathematics draws on science the way an artist draws on a muse**. Science reveals a real-world phenomenon—and the mathematician asks, how can we make this abstract? How can we generalize?

Take the idea of spatial dimension. Most of your ordinary objects—microwaves, teapots, housecats—are three-dimensional. That means they can be measured in three directions—length, width, and height.

Thus, any point in our three-dimensional world can be summarized with three numbers—call them, *x*, *y*, and *z*.

But the mathematician pushes further. What if we had four numbers? Or five? Or *n*? What would words like “volume” and “distance” come to mean? Can we imagine a six-dimensional sphere, and if so, what in blazes *is* it? Which of our 3D intuitions will carry over into higher dimensions, and which will break down?

For mathematicians, physical reality—that is, scientific reality—is nothing more or less than a source of inspiration. And like any inspired artist, mathematicians feel free to extrapolate and invent, to ask “What if?” and “How else?” and “Couldn’t we pretend…?” It doesn’t matter whether any of this higher-dimensional stuff really exists—it’s still a marvelous stroll to take your brain on.

Mathematics and science, then, aren’t like two members of the same species. They’re like two entirely different animals, sharing a lovely symbiosis. Math is like a parasite-eating bird, perched on the rhino of scientific reality.

Math gets nourished. Science solves a problem. Everybody wins.

The bird and rhino don’t share much in common, but they make a heck of a team.

Watch out, you’re treading in dangerous territory! But seriously, in theory math and science meet, and I mean that in both ways.

I think your post captures applied science, but as an aspiring theoretical physicist, I see mathematics as more than a toolbox–I see it as the framework of physical reality. Someday, the two will become one, and our minds will be blown.

That sounds fair – theoretical physics certainly dances along the border between the two (if such a border really exists).

Would you say, though, that it’s fair to describe theoretical physics as looking for the mathematical structures that correspond to the structure of physical reality? Because mathematicians are capable of dreaming up all sorts of frameworks and possibilities that seem to have little bearing on our actual material universe. In that case, theoretical physicists don’t see mathematics as a toolkit, but perhaps as a series of paint swatches, and they’re trying to perfectly match the color of the universe around them.

Well theoretical physicists tend to be a group of scientists who definitely view mathematics as more than a mere toolkit,well quite a lot of them that is and Arthur Phillip Dent here is certainly not alone.In fact Max Tegmark in his Mathematical Universe Hypothesis( MUH) goes as far as saying that every valid mathematical structure corresponds to a physical reality! (Given the fact that I am high school graduate only I can of course misunderstand his statement).I believe it is safe to say scientists,at least theoretical physicists do,have differing views of mathematics . I must also add, you are doing a spectacular job with the blog Ben.I am fan and follow it on regular bases.Keep up the good work.*thumbs up*

Yes, that sounds reasonable.

There are some strange connections between mathematics and physical reality. Consider the strange connection between quantum mechanics and the Riemann Hypothesis. I don’t understand it at all (yet), but it’s those mysteries that really pull me toward both physics and mathematics.

Agreed – who was it who wrote that essay ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’? I read it sometime last year and quite enjoyed it.

I believe Eugene Wigner was one of the first to publish on this topic, but there have been a number. Some of them marvel at the unreasonable effectiveness of mathematics, and others argue that it’s unreasonably ineffective for something that’s supposedly discovered rather than invented. As a result, I’ve always wanted to write a tongue-in-cheek essay on “The Unreasonable Ineffectiveness of Meta-mathematicians.”

Thinking here… whether a scientist believes that mathematics is discovered or invented might influence his/her beliefs on it being just a set of tools or something more. Personally, I think of numbers, algebra, calculus, and so on as mere tools, but they give a peek at something much deeper and much more profound. To me, that is mathematics–hidden there deep beneath the tools that many think of when referring to mathematics. I have yet to see it clearly, but everyone once in a while, I catch a glimpse of it, and that’s what drives me on.

I think the words ‘peek’ and ‘glimpse’ are very apt. All the language and symbolism of mathematics–our definitions, postulates, theorems, etc.–seem to be verbal signposts we use to make sense of that deeper, more profound ‘essential’ mathematics you’re talking about.

And Paul, you’re probably right about the contingency and temporariness of scientific truths, in contrast to mathematical ones. Euclid remains immortal; Ptolemy, once just as esteemed, is now only a historical artifact.

Oh, Star Trek… pfft.

$£$%£?!!

YOU COME TO MY BLOG AND YOU SAY ‘STAR TREK PFFT’?!?!

And there we go. (I’m an Old Republic Star Wars fan.)

Reblogged this on The Cardiac Muscle speaks and commented:

AWWW THIS IS JUST—-

Taking your brain for a walk: that sounds like ART!

Check out Lockhart’s Lament – one of the more eloquent defenses of mathematics as an art! (He draws analogies to music and painting.)

Thank you. That was a bit depressing, but clarified a lot of my own feelings about mathematics.

The day mathematical and physical constants are linked to each other will be the last day of all religions.

Oh, I think religion is flexible enough to accommodate pretty much any scientific development. Not that religion is the ONLY way to satisfy the human need for moral purpose, but it’s a very enduring one.

When math has nothing to do with physics, it is called pure. Mathematicians may know certainty. A physicist can only be certain of failure. It is not possible to prove that any theory cannot be wrong. It is easy to prove that the square root of two cannot be rational. Mathematicians deal in eternal verities. Physicists speculate. These subjects are not converging.

“Mathematicians may know certainty….”

You know, this kind of Hilbertian/Platonist assertion gets bandied about quite a bit. Sure you might “know certainty” (in the sense of, say, a conclusion axiomatically derived), but what they want isn’t that: it’s *understanding*.

See for instance the general math community’s reluctance over acceptance of the proof of the four-color theorem, of the sphere-packing problem, of stuff they can’t get their heads around like Mochizuki’s proof of abc. For the first two examples I gave, we proved these theorems with computers; I don’t know how more certain we can get than that (especially since they were independently verified). Why the hesitation? Because *we* humans don’t understand what’s going on. (Same thing for e.g. chess: see Stiller’s monsters here http://timkr.home.xs4all.nl/chess/perfect.htm)

See how Thurston (of geometrization conjecture fame) talks about the nature of proof in math (http://arxiv.org/pdf/math.ho/9404236.pdf), Cathy O’Neil’s argument that abc “hasn’t been proven” (http://mathbabe.org/2012/11/14/the-abc-conjecture-has-not-been-proved/), Reuben Hersh’s argument that “mathematics is fallible” (http://www.bgc-jena.mpg.de/~bsmolny/pmwiki/uploads/DasJahr2007/test1AlephBibNo1236.pdf), the last one developed in full in Lakatos’ (sadly unfinished) book “Proofs and Refutations” – worth getting for the first chapter alone.

The last link I gave actually has *eighteen* essays written by various authors from different backgrounds all trying to answer the question “what is mathematics?”. You’ll agree with me that dismissing them on Platonist grounds would be a bit naive (:

To be fair, I was working purely from memory so I might have mixed up my examples here and there. Feel free to point them out if so!

Things are never so cut and dry. Just because I study Chemistry doesn’t mean I think math is shallow, uninteresting, or limited to how I use it.

That’s very fair! 400 words and a few cartoons are never going to capture the full complexity of an issue like this.

I never meant to characterize the personal views of individual researchers, for what it’s worth – just to generalize about the role of math in science research, and vice versa.

That’s a good point. I was probably taking a few cartoons as much more serious an argument than it was.

those are my favoret subject

I do love a symbiotic relationship!!

Oh… and congrats on getting “Pressed”!!!

Stumbled upon your blog from twitter, absolutely loved it! “Taking your brain for a walk..” I hope you don’t mind me using that in my talks for kids.

Thanks – glad you’re enjoying the blog! Feel free to make use of anything you find helpful.

I love your humorous takes on education and learning. As a teacher for many years – and student forever – classroom stories have always fascinated me. I will Follow you starting today. On my WordPress blog – mikeandberg.com – you can read many blogs of humor (mostly) about learning experiences, from thesaurus use to bygone days of the common pencil to art education. I hope you’ll enjoy perusing my site and stories!

Yeah, they both are like a good example of mutualism to begin with. 😛

*sees maths, runs away from it* As a zoologist, I agree with this article. It was witty and hilarious, but oddly accurate.

Science and maths go hand-in-hand. They are both incomplete without each other.

I love all the doodles in your posts. Do you draw them yourself?

For more complications… this is the same as comparing Pure Math and Applied Math, they see “Math” differently. Applied Math is like Science in this article.