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Comment by mjburgess | Hacker News Reader

That's just "Exist as physical objects exist"

Ie., you're asking someone to find you an object with spatio-temporal properties which, as an object, does not have spatio-temporal properties.

Likewise you could ask, "find me a smile?!!" and then deny every example was a smile because "it was only a face smiling!".

"Existence" is defined relative to a context in which it can be evaluated, and it -- in general -- does not imply any particular properties.

"To Exit" in general is "to be found in some domain"

2 can be found in the domain of numbers

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This reminds me of my Italian grammar teacher telling me that nouns can be broken into two categories: ‘concrete’ and ‘abstract’ (for example, ‘chair’ and ‘love’ respectively). Then came a totally mind bending digression for my eight-year-old mind when she mumbled “well actually all nouns are abstract because there’s no such thing as a chair divorced from reality, there’s nobly instantiations of these abstract concepts we consider close enough to be concrete”. That totally blew my mind. Together with reading The Computational Beauty Of Nature” when it first came out nine years ago those were the two key experiences that drove me towards choosing to get a degree in mathematics.

  > all nouns are abstract because there’s no such thing as a chair divorced from reality,
  > there’s nobly instantiations of these abstract concepts we consider close enough to be
  > concrete

Very Plato for an 8 year old.

https://en.wikipedia.org/wiki/Prat%C4%ABtyasamutp%C4%81da

Agree. Maybe it is more helpful to think of numbers as adjectives than nouns. Oneness plus twoness equals threeness. The part of speech is sort of irrelevant but it obviates a debate over whether or not instantiation of the unattached idea of a number is important, while still allowing math to occur.

Yes, I could have really tortured my kids and said, you can't even show me a "ball", because it will always be a specific ball and not the archetypal concept of "ball".

I probably should, it'd be fun and they're a bit older than when I pulled the "one" game.

Make sure you say "show me ball" instead of "a ball"...

The "technical" term amongst philosophers is "ontological category":

http://blog.rongarret.info/2015/02/31-flavors-of-ontology.ht...

I believe that in fact the numbers do exist as physical objects exist. Just in their own universe.

When we reason about numbers, we're embedding a representation of a universe of numbers into our universe.

What we call "physical existence" could is also probably just be "math all the way down". Just not in such a way that the number two per se can be an entity for us to behold.

I believe that in fact the numbers do exist as physical objects exist. Just in their own universe.

How is it any different than believing that unicorns exist as physical objects, just in their own universe?

What we call "physical existence" could...just be "math all the way down".

Or, math is also a number of simplified alternate views of reality of varying degrees of "applicability." (Where "applicability" has to do with predictive power over reality.)

> How is it any different than believing that unicorns exist as physical objects, just in their own universe?

Good question, but I think there might be a difference. Suppose, in a galaxy far, far away, there are aliens. They probably don't have the concept of unicorns. However, they probably have discovered numbers, pi, Pythagoras theorem, e^(i pi) + 1 = 0, etc.

So, does the fact that something is discovered (rather than invented), and can be discovered many times independently in identical fashion, confer it some sort of elevated ontological status?

However, they probably have discovered numbers, pi, Pythagoras theorem, e^(i pi) + 1 = 0, etc.

We can only be confident of that if they're in this universe. Also, this is only a conjecture. It's just something some of us imagine would be true. You can't elevate such an idea to the level of empirical data.

So, does the fact that something is discovered (rather than invented), and can be discovered many times independently in identical fashion, confer it some sort of elevated ontological status?

Unicorns. Kirin. Narwhals. Rhinos. So you have two independent origins in culture for such a creature, plus two appearances of something similar in nature. Does that confer it some sort of elevated ontological status?

> Unicorns. Kirin. Narwhals. Rhinos. So you have two independent origins in culture for such a creature, plus two appearances of something similar in nature.

It's not at all clear that the cultural constructs are independent of each other or real world creatures. Particularly in the Western case, some of the earliest Greek accounts of the unicorn as well as Marco Polo’s account confirming its existence and difference from the contemporary evolution of the image stem from the distant East (from the European perspective) and clearly are describing a rhinoceros. They are also very similar to accounts of the Kirin.

Kirin is giraffe and upset Confucius as it should not appear in his chaotic time.

Some accounts of Kirin, yes, appear quite likely to be giraffes, others are nearly identical to the Western accounts of the monoceros of distant India (one of the original of the unicorn legend) and are almost certainly accounts of rhinoceros.

I would sadly have to say no. Because in some completely different universe unlike ours, the aliens have no concept of, say, "hemoglobin" because there isn't matter like ours: no atoms, no iron, no carbon. Unlike numbers, hemoglobin and unicorns contain a kind of arbitrary complexity. Yet one is definitely real. Some real entities are far removed from elementary math (even though they may ultimately rest upon it, many levels down). The question is: are unicorns bunk, or are they like hemoglobin.

I'm not claiming that if something exists (haemoglobin), we must have a notion of it and true statements about it in every universe.

Rather: if we have a notion of something and statements about it in every universe, then it must exist. However, come to think of it, that's bonkers.

Still, it's remarkable that mathematics allows us to make reproducible true statements about fictional objects.

> They probably don't have the concept of unicorns.

What would happen to your epistemology if every alien civilization has unicorns but some lack π?

> every alien civilization has unicorns

I knew it, Unicorns are real!!11!

I am not sure I can conceive of aliens that don't have pi (how can they have cylindrical rockets?).

Anyway, maybe we should let the professionals deal with this.

https://plato.stanford.edu/entries/platonism-mathematics/

https://plato.stanford.edu/entries/intuitionism/

https://plato.stanford.edu/entries/mathphil-indis/

https://plato.stanford.edu/entries/nominalism-mathematics/

https://plato.stanford.edu/entries/philosophy-mathematics/

> How is it any different than believing that unicorns exist as physical objects, just in their own universe?

We have very well-developed representations of the number universe that we can embed into our own universe. These things are governed by precise axioms. By contrast, we don't have such well-developed "unicorn universe".

(For starters, what is the definition of "unicorn"? If it's just a horse with a horn, then such a thing is plausible with genetic engineering in our universe.) As we endow the unicorn with additional properties, then it becomes less and less clear that there exists a universe where such a thing can exists other than as an imaginary being. If the concept of a unicorn could be axiomatized, then it could constitute a universe.

> Or, math is also a number of simplified alternate views of reality.

Well, that's the math that we do; not the math that we (possibly) are.

Indeed. The current better known attempts at a "unicorn universe" have problems with power scaling and the square-cube law. https://www.reddit.com/r/mylittlepony/comments/mrs81/accordi...

Do you have a good definition of what constitutes a "well-developed" universe?

>Do you have a good definition of what constitutes a "well-developed" universe?

An easy definition would be one where everyone agrees on the exact mechanics and sequence of states. The fibonacci sequence is well-developed: it has an exact definition that every competent simulator can simulate the same way. If I say my favorite number is the 1000th item in the sequence (starting with 0, 1, 1 as items #1, #2, #3), then everyone can calculate the 1000th item the same way.

If we all try to talk about what happens in the "unicorn universe", everyone has their own different ideas about that. There's no unambiguous rules for how the unicorn universe is calculated. Everyone has a different "unicorn universe" in their heads, and each of those universes operates on our own mind's whims rather than any rules we can describe to each other.

If we all try to talk about what happens in the "unicorn universe", everyone has their own different ideas about that.

Homo sapiens has this issue with this universe. Even if you subset that group to just scientists, there are still differences of opinion.

Reality itself is well-defined in that it acts consistently with itself. We as humans don't have a well-defined embedding of this universe into itself. We would need to know the complete underlying rules of physics for that.

That other "universe" is just patterns embedded in our brains. That's all you need.

You could likewise argue that sounds exist in a different universe, but are only "embedded" into the air using pressure waves, but why would you ever think you needed a different universe of sounds when you've already got pressure waves in air? I mean... what's "not enough" about that for you?

>What we call "physical existence" could is also probably just be "math all the way down".

This is a needless diversion. Mathematics is the study of structure. Physical reality has structure, and very well could "be structure" in some equivalent sense, but to say it is math is just abject poetic nonsense. This change of language would literally make no significant difference to anybody about anything. You may as well be declaring the world to be flat and asking everybody else to go and change all our language to make it true.

> abject poetic nonsense.

No it isn't! Either it is math all the way down, or else it isn't.

If it isn't, then what is at the bottom?

So this is actually an existential question.

Suppose we reach a final description of reality that is entirely accurate (no longer an approximation that fails to account for some structures and processes).

Won't that description consist of nothing but math?

Then, if it is complete, how does that description account for the fact that reality isn't math? If there is a difference between reality and math, can't that be described with math, so then it goes away?

If you want math to just be a description of a structure that itself isn't math, then the structure has to have attributes that are not covered by the math. The math is then using a substitute structure with fewer properties as a proxy for that structure. Once all the attributes are rolled into the math, there ceases to be a difference between that target structure and the math; it is that mathematical object. To prove otherwise requires one to name the difference. If two entities are completely described and found to be the same, they are just one entity.

> You may as well be declaring the world to be flat.

It is testable that the world isn't flat.

My guess is that eventually, if we manage to ever reach that point of total knowledge, there are certain properties of the universe that exist for no reason; they just are. Whether that's at the level of today's fundamental particles/qft or much deeper is anyone's guess.

Example: space-time exists in n(>=4) dimensions, there are several quantum fields that exist in that space-time and evolve/interact according to the equations of qft, gravity has some mathematical description (better than the ones we have now). And that's it, there's nothing else for us to know. Of course, we can never know for certain whether we've bottomed-out our description of reality, so physicists will always have some hope.

Now, if the universe is described entirely by mathematics, bootstrapped by a few base cases (the existence of fields/particles/space-time/strings/etc), I'd be comfortable with that. I'm not sure what a truly satisfying understanding of reality would look like beyond that.

> there are certain properties of the universe that exist for no reason; they just are.

But we have that in math and logic: axioms!

> Of course, we can never know for certain whether we've bottomed-out our description of reality

That we do not know is a fact; but is it a fact that we can never know?

What if one fine day we do know?

> But we have that in math and logic: axioms!

Yes and no. Axioms are chosen such that the math that is based on them makes sense to us (because the math reflects what we'd expect in the real world) and isn't self-contradictory.

So they do exist for a reason.

> That we do not know is a fact; but is it a fact that we can never know?

What if our universe actually has randomness that appears to come from "outside the system", i.e. there is really no way to predict the outcome of certain things from inside the system.

Picture you're sitting in a box trying to play Mikado. In theory you should be able to perfectly predict the outcome of every move you make - except there's a stranger shaking the box you're sitting in at truly random intervals.

For our reality this would mean that there's no way for us to develop an understanding of certain things and predict their behavior.

So essentially the end of the line for all sciences except theology and philosophy.

This possibility makes me somewhat uncomfortable.

On the other hand maybe everything is perfectly deterministic within our universe.

>Suppose we reach a final description of reality that is entirely accurate (no longer an approximation that fails to account for some structures and processes).

> Won't that description consist of nothing but math?

Math is just words. Any description of anything can be reduced to math if you assume the proper math will be developed to describe it. But the same could be said of any subject. You could say the universe is Mandarin Chinese if you find a perfect description in that language. You could say the universe is binary code if you found a perfect description of it in binary. You could say the universe is tree bark if you found a perfect description of it in tree bark.

You're saying nothing useful. You're just confusing the map for the territory. (And arguably, you will never find a complete description of the entire universe that fits in a network of human brains, so the universe can't be math for that reason alone.)

> You're just confusing the map for the territory.

I am not at all! We do math with symbols and diagrams, but those symbols and diagrams are about something: they are maps, and there is a territory that they are about. When I say "math", I include the concepts, not just the symbolic, graphical and other representations.

So what I'm saying is that if/when mathematics provides a complete map to that territory which is "the world", that territory must then be made of exactly the same stuff as other described-by-math territory-entities like "ellipse", "pi", "finite field", "Hilbert space", ... the map of mathematics doesn't describe any other kind of territory.

I'm not saying that the symbols and pictures used to carry out math (the "maps") are the world.

> you will never find a complete description of the entire universe that fits in a network of human brains

The description of the state of it as it is before us is monstrously large, but the initial conditions and rules might not be. E.g. the Mandelbrot set is just iterating on z^2 + c for various c. To describe isn't necessarily to evaluate.

The concepts are also just symbols, just in different media. They are also a map.

>if/when mathematics provides a complete map to that territory which is "the world",

Then it will encode all the information that exists in the universe. Which will not fit in your head, and thus will not be mathematics. Just because a territory is trivially a map of itself doesn't mean it is useful to call it a map.

>The description of the state of it as it is before us is monstrously large, but the initial conditions and rules might not be.

Reality isn't just the initial rules. It has state. That state must be described for you to have a complete description of reality. Yes, we could mathematically describe the complete laws of nature. But reality is necessarily more than just that, because the state is really damn relevant. You can't just ignore it because it doesn't fit your argument.

>Reality isn't just the initial rules. It has state. That state must be described for you to have a complete description of reality.

If reality is deterministic, then all the state is derivable from the rules and the initial state, and it's possible we could figure out the rules and initial state and become relatively sure of them. (We probably wouldn't be able to calculate the future, since evaluating the rules within reality probably necessarily takes more steps than reality takes during that time. But we might be able to calculate enough to see that it matches up with what we think the early universe must have looked like. We might notice that even the smallest of tweaks to the rules or initial state creates an avalanche of changes that doesn't line up with what we believe the early universe looked like, so we can be sure there are no free parameters.) It's fully possible that the rules and initial state could fit in someone's head. They might be ridiculously simple, like Conway's game of life in a few more dimensions.

(If you think "reality can't be deterministic because true randomness or quantum events exist", well you can fit true randomness into a deterministic theory by having the world branch of every random event outcome. The Many-Worlds Interpretation could be viewed as doing something like this for quantum events; it's a deterministic theory.)

Even if reality is deterministic, you still have to know which state to derive. And that process will have to account for all of the same information.

The strange thing about mathematics, as opposed to other noncorporeal ideas, is that it is nearly uniquely suited to being transmitted between brains. My idea of the number "three" and your idea the number "three" are probably very similar. This sort sensation that they have an independent existence leads people to the Platonic notion of a mathematical "realm" or "universe".

(personally I think mathematics exists in our brains, but also and crucially in the interactions between our brains, since a proof is never accepted until it is reviewed and so on. A proof that's just in my brain is no proof at all, even if it happens to be correct)

You could say that about most language we use: it's meant for transmitting thoughts between human brains. Why would it be bad at it?

Besides, there are plenty of mathematical concepts which are hard to understand. I don't see any uniqueness to math here.

Yeah, mathematical objects like numbers are concepts in our brains, or imaginary objects. Numbers in particular are imaginary objects used for counting. You can set up an isomorphism between these numbers and real-world objects, e.g., to count them.

You may be interested in: https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...

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