Is Math a Feature of the Universe or a Feature of Human Creation? | Idea Channel | PBS

Math is invisible. Unlike physics, chemistry, and biology we can't see it, smell it, or even directly observe it in the universe. And so that has made a lot of really smart people ask, does it actually even EXIST?!?! Similar to the tree falling in the forest, there are people who believe that if no person existed to count, math wouldn't be around . .at ALL!!!! But is this true? Do we live in a mathless universe? Or if math is a real entity that exists, are there formulas and mathematical concepts out there in the universe that are undiscovered? Or is it all fiction? Whew!! So many questions, so many theories... watch the episode and let us know what you think!

All Time 10 Videos:

Episode Links:

Weezy Waiter's "The Good Stuff"

Awesome Math photos from Nikki Graziano

Further Reading for the "Online/Offline" Episode:
Nathan Jurgenson:

EA and Guns Article

Sources:
Eugene Wigner

Velocity of an Unladen Swallow

Alain Badiou
Briefings on Existence

Lakoff / Nunez
Where Does Math Comes From?

Mark Colyvan
An Introduction to the Philosophy of Mathematics

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Music:
Level 5: Room for the Homeless
Binarpilot
Clockwork - Titan (geometry remix)

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[austinwilliams]

I have a graduate degree in mathematics, and I think this is a fascinating question. Here's my take on this.

Mathematical objects are, fascinatingly, both *objective* and *abstract*. That is an exceedingly rare pair of properties for a thing to have.

Everyone (including mathematicians like myself) will agree that mathematical objects are *abstract* (that is, they do not have a "physical" or "material" existence). And for many people (especially anti-realists) the conversations ends there.

But if you want to know why mathematicians tend to be mathematical realists, it is because we become deeply familiar with the *objectiveness* of mathematics. You see, unlike most other abstract things, mathematical objects do not bend *one iota* to a thinker's will. They may exist entirely in our mind, but they are not subjective objects. We cannot shape or carve them. They do not bend to our opinions or preferences.

Prime numbers, for example, have certain properties that arise from their definition itself. Any mind that springs into existence anywhere in any universe will (if they are fortunate enough to “discover” or “invent” prime numbers to begin with) would agree on the properties of the primes. The properties are *inherent* in the objects themselves, and these properties are the same for all observers.

Indeed, any mind that ever comes into existence in any universe would agree on *every* property of prime numbers, circles, squares, etc.

It is this property of objectivity that imparts on the mathematician the feeling that these objects exist "on their own". It is from this position that the mathematical realist argues that mathematical objects "actually exist" — even while readily admitting that they have no physical or material substance. It is why mathematicians describe the action as “discovery” rather than “creation”.

People *without* that deep direct experience of the objectiveness of mathematical objects (most of my students, for example) often have the sense that the "rules" of mathematics are somehow arbitrary. As if things are the way things are because a bunch of professors decided it was going to be that way -- as if they could have just as easily decided it was going to be some other way instead.

But such is not the case. These objects are they way they are with complete indifference to the way we feel about them.

Does this objectivity imply that mathematical objects “exist”? Well that depends entirely on how we define the word “exist”. The way the word is commonly defined is not strict enough for us to decide either way.

The anti-realist would say “the abstractness implies it doesn’t exist”.

The realist would say “the objectiveness implies it does exist”.

I say “mathematics is abstract and objective”, and I ask myself “what does it mean for thing to exist?”.

[CampingforCool]

Math is "real" in the same sense that language is "real". When you say the word "tree", there is no actual tree. The word is simply an abstract concept used to describe something real. It's the same with math. Math helps us accurately make sense of reality.

[CerberusKnox]

I think the INSPIRATION for math exists in real life. Angles, shapes, values, they can be observed throughout the universe. But we as humans decided they were angles, shapes, and values in the first place. Before humans existed, things didn't really have "value". They were just there. Humans, in an attempt to organize what they observed, decided that an apple and another apple were two apples.

[paulbeach]

When I was 16 I asked my Maths teacher "Is Maths something inherent in reality or just something we impose on the world?". He looked at me and said "Get back to work Beach". I ended up doing a degree in Philosophy while my Maths declined :)

[noidea]

Math is a method of thinking. It's as real as thoughts are.

[arturoarmendariz]

It's so nostalgic watching a video with old memes <3

[MrDreadpiratelynx]

Since you can't even prove that the Universe exists independent of your own perception of it, how could you ever prove that any piece of it does?

[Wave_Commander]

Math is the extrapolation of logical reasoning, founded in axioms and relationships. It's a study of how far can you push logic and still produce useful results.

[luisoncpp]

Let's be clear, in math we have mainly two things:
-Definitions
-Theorems
We invent definitions with abstract logical rules.
We discover the theorems, that are logical consequences of the rules we invented.
So, one part of the math is an invention and a language(the definitions) and other one is a discovery(the theorems).
Many things in the real world follow our definitions up to certain point in consequence they follow our theorems up to certain point. The fact is that with math we can discover universally valid facts that can explain things that had happened and predict things that still don't happen.
Does math really "exists"? I think that question doesn't have any sense, the concept of existence is not well defined and should not be used for this purpose.

[Necroskull]

Math is human logic in its purest form, and numbers are the language we use to convey math.

[Garbaz]

The main problem with this discussion is that most people don't have any clue what math is based on.
In mathematics we call this basis our "axioms". Axioms are facts we accept/assume to be true and use to deduce more mathematical facts.
The axioms for the average person are usually what they learned at school or, for those who actually listened that one time the teacher proved Pythagoras' theorem, what they learned at primary school. Namely stuff like "1+1=2" or "The square root of two has infinite decimal places".
In practice mathematicians aren't that different, but in theory, all math is based on some very abstract but basic principles.
Something like "1+1 = 2" might seem like good candidate for an axiom: Everybody would accept that it's true and it seems very basic. But for an axiom to be viable as a basis for *all* math, it has to have more to it than just being accepted by everyone. For example: If you accept addition, you can easily establish multiplication as repeated addition, or subtraction as reversed addition, but if you try to prove something about for example set theory, you will find it to be impossible to do so with this assumption ("Addition").
So after thousands of years of mathematical progress without a well defined basis, in the early 20th century mathematicians decided to clean the whole system up by defining these very basic axioms and actually proving trivial stuff like addition to "be true".
They are like the wooden blocks kids played with before smartphones were invented: You can do a lot with them, but after a while all possibilities are exhausted. You could build an even higher tower or an even longer bridge, but you can't build anything actually new out of them like for example a bowl.
To come back to the original question, continuing this analogy:
When a kid plays with its blocks, it might discover that it can place one block on two others and thereby build a bridge. Of course this kid weren't the first to build a bridge, but it _discovered_, without ever seeing one before, that it can create what we call a bridge out of its toys.
Mathematicians do essentially the same thing: They use mathematical building blocks and play around with them until they find some new combination.
The question is: Does the constellation of the bricks, the "bridge", already exist as an abstract concept, or does a child have to exist and create it for the "One block on two others" concept to exist?
If for example in absence of any conscious being one block happens to fall onto two others, thereby forming what our kid _invented_, we still get the same result *without* anybody inventing it. There isn't anybody there to call it "Bridge", but it exists the same way an undiscovered species or unknown chemical element exists.
The same way you can ask whether multiplication exists in a universe of addition. Maybe nobody thought about actually giving it a name (yet), but still, when repeatedly adding e.g. dinosaurs to a population, you are doing the same thing as we are when we are multiplying numbers.
So far it might seem like mathematical facts "exist" and are there to be discovered, since even without anybody there to think about them, they still apply.
Here is where the problem lies: Where do the wooden blocks come from? The bridge is made out of blocks, but what are the blocks made of?
You can break the block down into molecules, the molecules into atoms, the atoms into protons, neutrons and electrons, those into quantum particles, the quantum particles into ???.
Based on our current understanding of physics, quantum particles are the absolute basic building blocks of our universe and *everything* is made out of there.
Applying this concept to mathematics we can break down every theory, all formulas and any mathematical concept into the concepts they are based on, and in turn break those down, and so on until we reach our absolute basic building blocks, our axioms.
So in the end, mathematical concepts exist in our mathematical universe, which is defined by our mathematical capabilities / axioms, the same way that objects exist in our physical universe.
When a scientist discovers an unnatural chemical element by *producing* it via combining existing chemicals or when a physicist lets particles collide to produce new ones, did they really discover something or did they invent it?
How can you draw a distinction between those "discoveries" and supposed "inventions" in mathematics? You combine existing objects to produce something new.
Thereby, the real question is: What is the difference between invention and discovery?
You would call finding out that rubbing a balloon on your hair produces electricity a discovery, but creating an iPhone an invention. When Edison observed that letting current flow through a wire produces light, did he invent or discover the light bulb?
It is hard to draw a line between invention and discovery.
Thereby this whole discussion is interesting, but it is impossible to give a definitive answer, making it a slightly pointless endeavour.

[LunatixPLays]

It’s 2022 and I’m still hoping this channel makes a comeback…

[NoConsequenc]

I don't understand the question. The human brain is in the universe, part of the universe. To ask if math comes from the universe or the brain is to give no choice in answers...

[KarachoBolzen]

"Did quantum mechanics exist before we observed it?" My brain just broke.

[DronePunkGAME]

If I say math doesn't exist to my math teacher, does that mean I can get out of tests?

[cienciabit]

Math is both a feature of the Universe and a Human Creation, because we humans are part of the universe and even we create it with our conscience (and this conscience is, of course, part of the universe).

[slugfiller]

Math does not exist "out in the universe". At the same time, it does not simply "exist in our brain". It is actually in a layer that is beyond the universe. Math is not in the universe, the universe is in Math.

This may sound strange, awfully philosophical, or perhaps even meaningless, until you understand model theory.

The idea behind model theory is that mathematical rules can "map" into real-world concepts. Like, similar to the example already given, 2+2=4 can map into the real-world take 2 cats, add 2 more cats, get 4 cats. But the main premise of model theory is that you can map the same mathematical concept into several different elements. For example, 2+2 doesn't have to be about cats. In fact, it doesn't even have to be about physical objects. If you've waited 2 seconds, and immediately after you've waited 2 more seconds, then you've waited for 4 seconds.

In the eyes of model theory, then, math is about finding out what's common to different models. It starts by trying to find a minimal list of observable facts that a certain model has to fulfill in order to be "sufficiently similar" to other models. For example, in the case of addition, we require that we don't care about the order of the objects. For instance, if you put 2 black cats right of 2 white cats, then while you have 4 cats, this doesn't represent the fact that you have, right to left, black-black-white-white cats. Only if you decide this latter detail is unimportant, does the model fit. On the other hand, if color does matter, than taking the black cats first, and then putting the white cats right of them, changes the group of cats.
The next step is then to figure out what can be inferred based only on these observable facts, without relying on the observations themselves. For instance, 2+2=4 is actually a direct result of 1+1=2, 2+1=3, 3+1=4 and order of operations not mattering. The first 3 can be seen simply as definitions of 2, 3 and 4 (As what happens to the group when you add one more of the smallest group). They are giving names to concepts. The last one is the thing that must be true for a model to fit. Any aspect of this universe, or any other, that fits this requirement, would also fulfill 2+2=4.

The ultimate result of model theory lies in Godel's completeness theorem. It states that:

1. If something is true in a purely mathematical sense, then it is true for all models that satisfy the basic required observations. This means that math can be used to reason about the real universe. Another way to look at it, is that math can be used to show that certain types of things cannot exist, in any universe, simply because, in mathematical form, they would result in a contradiction.
A good example is that the laws of thermodynamics (conservation of energy) can't be violated without first violating one of the basic laws of mechanics (e.g. every action has an equal and opposite reaction) that mathematically cause them. So long as the basic observations hold, the conclusions must also hold. Conversely, if an observation shows that the conclusion was violated, it implies that one of the basic rules on which it was based was violated as well.

2. If something is true for all models, then there is a finite mathematical proof for it. Or, conversely, if there is no finite mathematical proof for something, then there is a theoretical model in which it is not true. While this fact may not directly relate to this universe, as the "model in which it is not true" may simply be a theoretical universe that doesn't actually exist, it does give us a way to limit the things that must be absolutely true for all possible universes. All of those must have FINITE proof, that is, the proof can be written using mathematical symbols on a finite number of pages.
At the same time, every time mathematicians think they've found a model that could only exist in theory, the universe has shown that if it's mathematically possible, then there's some aspect of the universe in which it physically exists. That's not to say that this is always the case, simply that mathematicians are still in search of the case in which it isn't.

[SMgrimbldoo]

Math is a human invention, but it is not "made up." It is our way of explaining the already existing things around us, which is pretty much the definition of spoken language. Essentially, math is a language.
On a side note, religion is not the opposite of science, in fact, it's not even related to science. Science is the explanation of the existence while religion is the belief, or disbelief, that a higher power created existence, simply the belief in higher powers or the supernatural, or an interest that is held and pursued at a supreme importance. By this definition, science can even be a religion if a person holds it to be important enough.

[illustriouschin]

the constant meme references is too distracting.

[ToasterShow]

"In the Math we currently there is - More Mathe" Oooh no and I thought I could get rid of it. But now I know better.