 This episode might be a little complex, you might want to square the last half of it just to be on the safe side. In 1623, Galileo Galilei published a book called Il Sagittore, or The Experimenter, which contained a philosophy about how one should approach the natural sciences, like physics or astronomy. Philosophy is written in this grand book, The Universe, which stands continually open before our eyes, but cannot be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles, and other geometric figures, without which it is impossible to humanly understand a word. Because you've had some sort of modern education, your first reaction might be some form of, well, duh. Mathematics is just how you do science, right? Well, math has been associated with the sciences for a long time. The Galileo was actually one of the first to put his foot down and say unequivocally the principles by which the universe operates are fundamentally mathematical in nature. He's not just claiming that mathematics is a useful tool for figuring out how the universe works. He's claiming that the underlying rules that make things tick are inherently mathematical, that the universe itself is basically written in math. That's less obvious. Do the mathematical principles that we learn actually have some sort of existence in the universe itself, which mathematicians just stumble onto by thinking hard? Or are they purely mental constructs that we invent that just happen to show up a lot in the universe's operation? Is mathematics discovered or invented? That might sound like a strange question at first. It helps to understand why math is so categorically different from everything else that humans use to understand things. First, it's true that many of the things that mathematicians work on eventually find some sort of practical application. But if you ask them, they'd probably tell you that their work has nothing whatsoever to do with the real world. For example, the Bonak-Tarski theorem states that through some set theory juju, you can take a solid sphere, divide it into an infinite set of points, then disassemble and reassemble those points into two spheres identical with the original one. That's not just physically impossible. It doesn't seem to provide any useful insight into anything but weird mathematical constructs. It's what's known as pure math, math that's untainted by anything like application or reality. And it's not categorically different from any other sort of math. Second, mathematics is generally considered to be composed of analytic statements, statements which don't require any sort of external confirmation to be true or false. All science, whether it's biology or astrophysics, requires measurements and observations to evaluate the accuracy of theories. Their truth or falsehood is only established by testing the behavior of the universe and seeing how closely it matches their predictions. But I don't need to grab three avocados and show you that I can't physically divide them into equal groups larger than one to say something like 3 is prime and be right. I don't need to put error bars on a graph of y equals x or say something like 1 plus 1 equals 2 so far. Unlike scientific facts, mathematical facts seem to be necessarily true, even if they're only about made-up stuff like the Bonak-Tarski theorem. The problem is that while we're playing around in the pure thought of math land, we continue to find these imaginary things seemingly written into the behavior of the universe, exactly as Galileo suggested. In his aptly named essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, physicist and philosopher Eugene Wigner laid out several convincing examples of exactly the sort of phenomenon. The first works we know of that reference conic sections, like parabolas, are from a dude named Menachemis in ancient Greece, who invented these shapes because they made it easier to calculate the answers to a pure math puzzle that he was working on. But despite those pure math origins, conic sections show up everywhere. The movement of projectiles, harmonic oscillators, radar dishes, finance, even the shapes of some sea creature shells are best described by these imaginary figures. Isaac Newton subscribed to Galileo's ideology about the fundamentally mathematical nature of the universe. Despite the fact that the astronomy charts he was using were relatively inaccurate, he guessed that the numbers that they were describing were vaguely elliptical in nature. The equation that he came up with, based on that guess, referencing these ancient mathematical curiosities, was absurdly accurate, down to a ten-thousandth of a percent. And he got it just by assuming that whatever made the planets move, it did so according to a simple mathematical formula. That's not really a well-duh thing anymore. These relationships are incredibly complicated and very hard to grasp, unless you're using mathematics to explain and predict them. Again and again, scientists guess simple mathematical answers to complex phenomena and end up bang on the money. It's so common, but the large part of the support for many modern theories in physics, things like m-theory and supersymmetry, are largely based on their mathematical elegance. Theorists just expect the universe to behave in a mathematically simple and elegant fashion. But again, there's nothing categorically separating the pure math, like the Benektarski paradox, from the applied math that we use to calculate things like orbits. We're caught in kind of an awkward place here. On the one hand, mathematics seems like it's separate from the real world in some significant way. On the other hand, it seems to show up there an awful lot. Unlike most metaphysics, the fundamental nature of mathematics has some serious practical ramifications. For example, what we think about math has a lot to do with what we think about the objective truth of science. Almost all modern scientific theories use some sort of mathematical justification for their claims. If that mathematics isn't objectively true, then all scientific findings that use math would, at best, be useful imitations of what's actually going on. Attempts to communicate with aliens, including the golden record on the Voyager probe and the Arecibo message, invariably assume that any intelligent life would be able to recognize something like pulses of prime numbers and interpret them more or less the way that we would. If mathematics is just a function of the human mind, we might be wasting our time in that regard. Philosophers have developed several different approaches to the question, but as far as we can tell, there's no real way of knowing which, if any of them, are right, and there are significant problems or objections to each. Lowercase P. Platonism is more or less derived from the philosophy of Plato, but has evolved a great deal since his theory of forms. According to Platonists, we discover mathematical entities and mathematical truths because they literally exist, and they're literally true. When you say something like 3 is prime, for a Platonist you are making a true statement about a real abstract thing called 3. Now, I'm not talking about the numeral 3 here, this little squiggle that we write on a piece of paper to reference 3-ness. I'm talking about the 3-ness itself, the thing that we recognize echoes of in groups of three things. This is actually how a lot of mathematicians think while they're working, as though they're searching for some answer that's already there. But, if abstract objects like numbers actually exist, they have to be pretty weird. If 3 has always been prime, even before anyone discovered it was, that means that it had to have some sort of existence outside of a human mind. But it couldn't have a specific location, or a physical presence, or a color, or any properties at all besides eternal, unchanging 3-ness and all the mathematical properties that implies. That's really, really weird. Mathematicians might think like Platonists while they're working, but if you ask one if they believed in the eternal, non-spatial, non-temporal, non-causal, abstract entity of 3-ness, they'd probably look at you like you were crazy. On the opposite end of the spectrum, we have nominalism, Latin for name-ism. The view that what we're doing when we count things is just drawing an imaginary boundary around some part of the universe and giving that part of it a name. For nominalists, there isn't really any such thing as 3-ness. It's just a convenient label that we stamp on things, we decide are like this 3-idea that we've come up with. At first, this seems pretty simple. One might be relieved to dispense with the abstract objects of Platonism, but nominalism has some problems of its own. We generally accept scientific knowledge as being literally true, along with the existence of the concepts that it uses to help explain certain phenomena, stuff like the Higgs field or dark matter. If we're willing to accept the existence of something like gravity in the course of a scientific explanation for phenomena, why wouldn't we accept something like numbers, which are just as necessary? Now, these are just two very broad positions about mathematical realism or anti-realism. The full landscape of the metaphysics of mathematics has a huge number of very nuanced and interesting responses to the question about whether math is discovered or invented or neither or both. Some philosophers believe that mathematics is just the inherent structure of logically valid thought. Others believe that all mathematical statements have an implied disclaimer, like in the fiction of arithmetic. Still others believe that the question itself is some sort of meaningless word game, like what's the sound of one hand clapping? Needless to say, there's a whole lot more to this that I'm not telling you. If this subject is of any interest to you at all, please, please check the links in the description. But even with all this careful nuance thought on the subject, there's still something intriguing and almost magical about mathematics and its efficacy that's hard to explain, especially when I'm trying to figure out how much money I spent on steam this month. What do you think? Is mathematics discovered, a feature of the universe itself, or is it invented, an imaginary man-made tool that just happens to be really, really good for science? Does it even mean anything to ask? Please, leave a comment below and let me know what you think. Also, episode 100 is coming up and I'd love to answer some of your questions, either about me or the show or any of the topics that I've covered, so please feel free to leave those as well. Thank you very much for watching. Don't forget to blah blah subscribe, blah share, and don't stop thunking.