Is mathematics invented or discovered? In my experience, applied scientists often think of mathematics as a human invention, while actual mathematicians (with a few notable exceptions) feel sure that mathematics was always there to be discovered. (Of course, it’s sometimes hard to tell how much of this is genuine disagreement and how much is a language barrier.)
I’ve just finished reading an excellent book by Mario Livio which is entirely about the invention/discovery question, though he’s chosen the (somewhat unfortunate) title Is God a Mathematician? Much of the book is a lively romp through mathematical history, with a well chosen mix of biography and exposition. Although he parts company with them in the last chapter, Livio gives a more than fair hearing to the many great mathematicians who have insisted that they are discoverers, from Pythagoras through Galileo, G.H. Hardy, Kurt Godel, and the contemporary Fields Medalist Alain Connes (among others). Here, for example is Connes:
Take prime numbers, for example, which as far as I’m concerned, constitute a more stable reality than the physical reality that surrounds us. The working mathematician can be likened to an explorer who sets out to discover the world…We run up against a reality every bit as uncontestable as physical reality.
Readers of The Big Questions will know that I am entirely in Connes’s camp on this issue, for reason I’ll blog about later in the week. And as I’ve said, it seems that most mathematicians sit in this camp. But there are notable dissenters, including the great Sir Michael Atiyah, another Fields Medalist who I might well have included in my gallery of heroes. Despite my great admiration for Atiyah, I believe he’s wrong on this issue. But more fundamentally, I believe his primary argument proves exactly the opposite of what he thinks it does. Here is that argument (slightly condensed):
Any mathematician must sympathize with Connes. We all feel that the integers really exist in some abstract sense and the Platonic view is extremely seductive. But can we really defend it? It might seem that counting is really a primordial notion. But let us imagine that intelligence had resided, not in mankind, but in some vast solitary and isolated jellyfish, buried deep in the Pacific Ocean. It would have no experience of individual objects, only of the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count.
Here Atiyah has envisioned a world where the natural numbers get neither invented nor discovered. I’m not sure why that’s supposed to prove they don’t exist. On the contrary, it seems to me that quite unbeknownst to Atiyah’s Jellyfish, the earth would still have exactly one moon and exactly two magnetic poles, and two would still be twice one. Two would still be a prime number, and 1729 would still be the smallest number that is the sum of two cubes in two different ways.
Indeed, all these things were true back in the days when the world was populated by creatures who were unaware of them. It doesn’t matter for the argument whether those creatures are highly intelligent in other ways. In that sense, Atiyah’s Jellyfish is more like a Red Herring.
Or to put this another way: Atiyah says that an intelligent creature might be unaware that one plus one makes two. Sure—so might any creature. To me, this indicates that 1+1=2 is not an invention; it’s simply a truth. Can anyone explain why Atiyah thinks his story proves otherwise?
quick, before the others see! somewhere in your post is an 8 that should be a 9.
of course, it’s really a 9 regardless of what you say, which I suppose is the point.
Xan: It’s fixed now! Great catch; thank you.
Interesting point.
I am not a mathematician, nor am I someone who claims to know anything about the origin of anything.
But this post reminded me of a piece in Joseph LeDoux’s book, The emotional brain – that the function is universal (he was talking about the mind body problem, I have related the example here)
He says, for the human being, for a calculator (computer), the idea of adding say 1+1=2 is a matter of function irrespective of the hardware. (the argument that the machine is programmed by a human is alright, but the point is that it is not where the calculation is made that matters)
Similarly, a visual representation of an addition function is sufficient for most intelligent life forms and not exactly the numerical symbolism in the mind, which we prescribe to while counting/adding. (eg: one mother, two predators, one herd, et al)
Maybe humans have ‘discovered’ a better way to symbolize the numbers.I don’t know what to call it, I am not sure if our human language can encapsulate this thought. This is something similar to explaining quantum physics in brief.
Hi..What do you think of the work of Reuben Hersh (“What is mathematics really?” and “Descartes’ dream”)? Roughly, what he says is something like the below:
“1+1=2” is true, but so is “2+2 = 1”, modulo 3. You know that..
Counting comes naturally and a feeling for small numbers is biologically useful, but turning that into an abstract system of elements and operations is a creative act and involves choices.
It is guided by the uses we find. We could imagine a civilization inventing non-euclidean geometry before euclidean geometry. Leibniz’s analysis using infinitesimals could have won out over Newton’s approach.
Its more subtle than that, of course, and he makes a good point about arguments such as “the earth would still have exactly one moon and exactly two magnetic poles”
Re the point about the moon.
The point he makes is that while there are things which can be counted, there are no numbers out there. You need humans to count. (How physical laws work, I have no idea, but it is unlikely that an electron measures the distance to another one and computes what it should do; let alone measure the interval between it and all the other electrons in the Universe).
Things don’t come labelled with what sets they belong to: I could count my cat as one of several furry things (including a teddy bear) in my house, or as one of several four legged things in the house. I could count the moon among the satellites of the Earth, th emajor objects of the Solar System, or among the the objects of Romantic significance in my life.
All are human choices, and the counting is a human act..
Russell said that a natural number is the set of all sets that have a number of elements. The number 5 is the set of all sets that contain 5 objects (fingers on my left hand, cats that live in my house, keys on my key ring, etc.). Thus any item can be counted in multiple countings, depending on the sets in which it’s counted. The ignition key to my car is part of the set of 5 keys on my key ring, and the set of 4 keys to my car, and the set of 9 keys in my backpack right now.
autogen, are you saying your belief is that Mars didn’t have two moons until we counted them, or that humans didn’t have 5 fingers until we could count to 5? That seems almost creationist as a position.
The sense of paradox which the question provokes is false.
Numbers do not exist in the same sense that platypuses exist, as concrete objects in the physical world. If they are said to exist at all, it must be in another, entirely different, sense.
In other words, the number two exists as a mathematical object in our system of mathematics. The poles of the earth exist as physical objects. The fact that a rotating body must have two poles shows that there is some relationship between the laws of the physical world and our system of mathematics. However the relationship is contingent and must be determined empirically.
It sounds like Atiyah is saying that a lot of what we consider math, does no exist outside of our heads; it is an artifact of the way we see the world, so it’s a construct of the way our minds work. He seems to be arguing that we organize our observations of the world in certain ways that are peculiar to us, rather than inherent in the world, and that math is an expression of that organization that we impose on the world as part of our worldview.
Your example can be used to support his point. What does it mean that there’s one moon and two magnetic poles and that two is twice one? Are there twice as many magnetic poles on the earth as moons around the earth? That can only make sense once we’ve abstracted the numbers away from the things they represent, because in the real world, the relationship between “magnetic poles” (which aren’t even discrete things, they’re concepts) and moons of the earth, seems to have nothing in common with the relationship between the numbers 2 and 1.
Was air conditioning invented or discovered? The physical principles underlying air conditioning were always there, and we would not have air conditioning if the laws of nature were not amenable to it.
I’m not sure, but I’m kind of leaning toward “language question”.
Also, I hope whoever writes the next comment has a name starting with an “E”.
Atiyah’s jellyfish world is not, of course, a pure continuum. It contains at least one discrete thing: the jellyfish itself (and maybe also the continuum itself). So the jellyfish exists (=def ‘1’). So also does the set containing the jellyfish ( {jellyfish} =def ‘2’ ). So also does the set containing the set containing the jellyfish ( {{jellyfish}} =def ‘3’ ). And so on. I’d guess that a jellyfish of even moderate intelligence would recognize this, and proceed to discover the entirety of arithmetic.
Although having thought about it, perhaps the jellyfish would get stuck along the way when it required the Axiom of Choice. The self-evidence of AC seems to me depend on an extrapolation from our lived experiences of choosing one from a finite set of objects; maybe it wouldn’t be quite as self-evident to a being that had spent its existence in a pure continuum with no other entities.
AV, I think Ben gets it exactly right.. There are things there which can be counted, but those numbers themselves don’t exist out there. Even those sets/collections don’t exist except when we look at the world and decide to group some things together. When we see sheep grazing, we instinctively group them together and talk of the “sheep on the hill” but what exist are the sheep. The collection exists in our head.
I agree with Ben and autogen. Imagine that our moon split in half. Now how many moons would we have? We have two bodies orbiting Earth but exactly the same matter that was there when we said one. “Earth has one moon” is not true in the same way that “1+1=2” is, like Ben said. We can use numbers to represent the physical world but such representation is not an inherent part of a universe without human minds to think about it.
Wow! I’m quite surprised how many people believe that numbers exist only in our heads. I am firmly in the camp that numbers, and all of mathematics, exist independently of us, and in fact independently of our universe, as described in Max Tegmark’s “Mathematical Universe”.
I tend to agree with Ben, autogen, and Saber on this point. Numbers are mental constructs that we use to describe the world. In Atiyah’s example, these constructs don’t exist because the jellyfish haven’t created them. In this type of scenario, numbers may still exist, but only in the sense in which any potentially conceivable mental construct exists.
If in the future a more potent method of understanding the world is conceived that replaces the number system, what would that mean? In the 17th Century, a proponent of Phlogiston might have argued that Phlogiston always existed, but we now have a more advanced system of concepts.
In any case, Atiyah’s point was that counting is not clearly a primordial notion. At least in the excerpt provided, he wasn’t attempting to prove that numbers “exist” even if they are neither invented nor discovered. In fact, he clearly assumes that counting is a “notion.” It remains to be argued that numbers are not a notion, but are something else, and, if so, what.
I fell into the ‘numbers are abstractions’ way back in 5th grade when we learned to calculate Pi. We can never know the exact value for Pi, and just use an approximation that gives an answer with the accuracy needed, whether its 3.14, or 3.141592353589793.
The same could be said of counting apples, or moons. If I have two apples, does that mean both apples are identical? If I mash up 10 sets of 2 apples, will all sets have the same mass? You can even use that type of thinking on subatomic particles. If you have 10 protons, not all of them will be exactly the same, but we can abstractly call them all ‘protons’ and count them up.
Once I took higher math in college, I became even more convinced that math was an abstraction that we overlay on reality. Calculating the minimum number of stars in the observable universe, we come up with three to seven times ten to the twenty second power. I seriously doubt that the number of stars would add up neatly to a figure with 22 zeros, and since stars are constantly being born and dying, we come up with an estimate to work with. And how different is an estimate from an abstraction?
I’m not entirely convinced its possible to do mathematical research without being convinced that the reality is ‘out there’ somewhere. You are doing research: there is only one possible answer to the question you’re asking, and that answer would be the same whether you had asked the question, someone else had asked the question or no-one had asked the question. As for the point that 1.
As for “”1+1=2″ is true, but so is “2+2 = 1″, modulo 3. You know that..”. I have almost literally no idea what the point of that sentence is. Are you suggesting that there is some other part of the universe in which the remainder you get when dividing 4 by 3 is different to 1? That it would be possible for an intelligent creature to ask itself “what is 2+2 congruent to modulo 3?” and come up with an answer other than one.
As for the historical examples. Obviously what is *known* about mathematics is a cultural issue. As Steven said, for most of the history of the universe nothing was *known* about mathematics, but 4 was still congruent to 1 modulo 3.
im also in the ‘we create numbers’ camp. there may be two poles, but there is only one magnetic field. without some lodestone, even the existence of the field would be difficult to imagine.
on a side note, im under the (often mistaken) impression that a bodies magnetic field had to do with its composition and not its rotation. illuminate me on this?
dave, assuming there are no magnetic monopoles, where there’s a magnetic field, there must be moving charges (or changing electric field) to produce it. A stationary charge just produces electric fields. In many materials, there is plenty of “motion” in the relevant sense at the atomic level, so you can get magnetism that way; for the production of large magnetic fields, though, you generally move currents in a more macroscopic way.
The question: “Is mathematics discovered or invented?” is the same question as: “Does God exist or not.”. The human mind is constructed so as to come up with the two answers, “Yes” and “No”. When humans think deeper about such questions they come up with the third answer, “I do not know.”.
On even deeper thinking a better answer is: “I do not understand the question.”
I think an even better answer is: “I do not understand the question and we can show mathematically that you do not understand the question either.” Having Gödel and Tarski in the list would have helped with this answer.
The question then becomes: “Why do humans insist on asking question that they know they do not understand.”
If we think of consciousness as an emergent property of motile organisms that has the purpose of modeling its sensory inputs so it can predict the way that environment will evolve so it can then successful negotiate that environment in order to persist then and important attribute of that modeling engine is that it must not freeze up or grind to a halt on in input set.
This means that an organism so equipped when presented with a question that it does not understand naturally will come up with an answer. The more complex the organism the more complex the answer.
Even the moon is not a discrete, countable object if you sense it at the quantum level; it has a wave function. How about quantum energy levels, though? Might they be discrete and countable?