Does Math Exist?

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Memory plays tricks, but I think I remember this moment quite well. It was 2 years after my Baccalauréat. In the midst of a massive ingestion of math and physics in “prépa”, I could not help thinking about it all after classes. My thoughts that night were on the laws governing electric and magnetic fields. The so-called Maxwell’s equations. They involve a mathematical arsenal called “partial derivatives”, discovered by Newton and Leibniz in the seventeenth century.

Lost in my Maxwellian dreams, I started to wonder: “How is it that abstract mathematical tools developed over millennia are suddenly perfectly adapted to describe something real?”

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Yes, mathematics exists – as a human cultural practice.

I think, however, that Antoine is really looking at the questions “Do mathematical objects (such as numbers) exist?” And people disagree about that.

But we might just as well ask:

  • does existence exist?
  • is truth true?

My point, of course, is that people disagree over the meaning of “exist” and over the meaning of “true”.

So Antoine discusses two possibilities – mathematical platonism, and formalism. But that’s too narrow a choice. I’m a fictionalist, so I am neither a platonist nor a formalist.

The mathematician Kronecker famously said “God gave us the natural numbers; all else is the work of man.” And that’s clearly a rejection of platonism. Personally, I think Kronecker gave God too much credit. The natural numbers are also human inventions.

Concepts such as “exist” and “true” emerge from cultural practices. And that’s why it is hard to pin them down.

When I am doing mathematics, then numbers exist. I can look at an equation, and talk about the existence of solutions (possibly unknown) for that equation. And when I do that, I am taking for granted that numbers exist. But when I am not actually doing mathematics, but instead trying to explain how it works, then numbers do not exist. They are useful fictions.

If I am reading a Sherlock Holmes story, then I have entered the fictional world of Sherlock Holmes. And, inside that world, Sherlock Holmes exists. There would be no fun in reading a Sherlock Holmes story if, every moment, I was questioning whether Holmes exists. But once I leave that fictional domain, no Sherlock Holmes does not exist.

For me, it is much the same with numbers and other mathematical entities.

As to why mathematics is so useful. I commented on that in another thread:

Science is systematic, and mathematics is about systematicity.

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“How is it that abstract mathematical tools developed over millennia are suddenly perfectly adapted to describe something real?”

Even in my own very limited experience with higher mathematics, I regularly pondered this question.

For example, I was amazed that George Boole developed so extensively what came to be called Boolean Algebra long before there were any electronic logic circuits and circuit boards. And when I taught VLSI (Very Large Scale Integration), I explained the wonderfully practical Karnaugh maps (especially for reducing gate and IC chip counts)—which were developed in 1953 but which actually represented a rediscovery of Marquand Diagrams, published in 1881. Long before transistors and even the first diode, Marquand worked out the math which helps minimize electronic elements and power consumption.

Yes indeed. For centuries philosophers and mathematicians have debated whether mathematics exists on its own or is merely something which exists in the human mind.

The details of such debates are beyond my abilities—but I think it very cool that mathematicians keep working out amazing abstractions which decades later are applied to solving some “real life” problem.

If I heard correctly, the solution for Fermat’s Last Theorem required the application of some really really abstract mathematics which was not all that interesting to many—until it got applied to the marvelous proof. That’s also very cool.

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Being an academic, this is exactly the type of thing I talk about in defense of the liberal arts. Certainly a balance is needed between professional and liberal arts (I find the best degrees are a blend) but just because something lacks an immediate commercial value right now, doesn’t mean it’s useless or purely theoretical.

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While realizing that not everyone has the time and funding to take advantage of such options, my degrees in both the humanities and science served me well. Faculty experiences in both science and theology gave me an interesting career and I feel like I can understood both cultures—both private (secular as well as evangelical private universities) and public university environments. Peer-review applies in all of those tracks, obviously, though not everyone accepts that fact!

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Math doesn’t exist. instead the bible says God measured out the universe. So we can measure it. thus math. Math is however a human construction of this measurement and so doesn’t count.
its just a special case.

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The mathematical relationships preexisted man’s discovery of them, E.g., Euler didn’t invent the beautiful relationships in his eponymous identity – they were there all along.

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Agreed. But we did invent the number system in which that relationship shows up.

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I’m not so sure it wasn’t discovered, discovered to be the most practical system. (It ain’t Roman numerals! :slightly_smiling_face:) We invented the Arabic numerals labeling, maybe, but those are arbitrary. The geometric relationships in Euler’s formula certainly preexisted him.

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Keep in mind that Euler’s formula requires complex numbers. There are quite a few books out there that describe the complex numbers as an invented number system.

Once you had the integers, inventing the rationals was a reasonably natural thing to do (that is, it came naturally to mathematicians). And from there, inventing the real numbers seemed reasonably natural. But inventing the imaginary numbers (and thus the complex numbers) has long seemed to be an unnatural step, though it worked out very well.

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I always thought it was intuitive that the square root of -1 was i. :upside_down_face:

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