Comment by meroes | Hacker News Reader

Comment by meroes | Hacker News Reader

How math relates to the physical world is not at all clear. Like yes 2 apples clearly makes sense, but non physical quantities like 10^241 of anything start to break down the relationship.

It’s not the abstraction that gets me usually, it’s the lack of an explicit relationship between math and physics. I’m sorry but personally I am not satisfied by what I learned in my math and math/science classes on this. It started for me day 1 of high school physics and I’m now graduating in 1 month in undergrad applied mathematics.

I felt reinforced when I read that Plato clearly delineated the two “realms”. I was hopeful a single math teacher would discuss this idea for one lesson, but it never happened. Plato was a champion of math who recognized the division.

I’m not one of those disbelievers in complex numbers or anything like that, but if some model formulates precise physical scenarios, we should be able to have ample cross-over language. Even the explanation for why honeycombs have six sides: there is always some kind of fumble from going from the mathematical reason to the physical reason. It’s the paradigmatic example of mathematical explanations for physical phenomena, and between two heavily studied fields, and yet we fumble it continually. What is it about the mathematical hexagon which determines the physical honeycomb? These questions are at the height of philosophy of mathematics, e.g. Mark Colyvan, yet are still disputed and we act like it’s all so obvious. I get it works, but don’t tell me it’s not mysterious, because 6 years in now I’m not at all satisfied and have little confidence it will be mentioned in future math/science classes.

Can you elaborate on why you describe 10^241 as non-physical? Just because it's too large to describe anything physical?

Anyhow, Aristotle had another take on the problem of universals, and taking his view that every concept abstract or otherwise can only exist as part of our physical world (I'm not a philosopher, and there are variations on both themes that different people subscribe to), then your supposition would be nonsense in itself. Stated as a question: What is your reasoning for preferring a Plato-like model of the world vs. a Aristotle-like one?

(If it needs to be said: This is not a gotcha or anything - your opinion intrigued me, and I'm curious as to your reasoning)

Well, based on a quick Google, there are approximately 10^186 Planck-length volumes in the known universe. So 10^241 is not physical in the sense that there is no way to 'realize' a set of 10^241 things.

Another related notion is that there is no way to 'realize' an irrational 'number' -- sqrt(2) can be defined but we can never draw a length of sqrt(2).

It really depends what you consider 'real'. It is actually pretty straightforward to realize a representation of 10^241 (you just did). Why does that make it any less real than the digit 2, which represents two things, but is actually just itself one thing.

> Another related notion is that there is no way to 'realize' an irrational 'number' -- sqrt(2) can be defined but we can never draw a length of sqrt(2).

Again, it's unclear what anyone means by things like 'sqrt(2)'. Why is drawing a thing of length sqrt(2) any different than drawing a thing of length 1? If I draw a thing of length 1 and say it's a line of length 1, then why is that different than my drawing the same line, claiming its length is sort(2) and then pointing out that it's actually now impossible to mark where 1 would appear along its length?

> It really depends what you consider 'real'. It is actually pretty straightforward to realize a representation of 10^241 (you just did). Why does that make it any less real than the digit 2, which represents two things, but is actually just itself one thing.

One is a map, and the other is the territory. Both 'real' in some sense but a map without the territory feels less 'grounded' (pun?).

> Again, it's unclear what anyone means by things like 'sqrt(2)'. Why is drawing a thing of length sqrt(2) any different than drawing a thing of length 1? If I draw a thing of length 1 and say it's a line of length 1, then why is that different than my drawing the same line, claiming its length is sort(2) and then pointing out that it's actually now impossible to mark where 1 would appear along its length?

Perhaps a more precise way to describe the situation is that one can define one or the other as a base 'unit' but you can never get one from the other (they are 'incommensurate', as the greeks would say). Irrational numbers can be defined but not 'realized' (or 'constructed') in the same way that the rationals can.

> Irrational numbers can be defined but not 'realized' (or 'constructed') in the same way that the rationals can.

Once again, I'm not 100% sure what you're saying. If you have something that's of length 1, then you can easily construct the line with a ratio sqrt(2):1. Draw another line of length 1 (use compass and straightedge) at a 90 degree angle. Repeat 4 times until you have a square. Now draw the diagonal. You have successfully constructed the square root of 2 in your own setup.

There are numbers that cannot be constructed geometrically using only a compass and straightedge (e for example). However, again these are no less 'real'. Drawing lines is not the only measure of real. There are other methods. In computing, we often say a number is computable if you can define a function that, given any rational number can tell you if the number it represents is greater than or equal to the rational number. The square root of two and e and pi, etc are easily representable this way. There are some numbers that are not, and perhaps these can truly be said to not exist.

However, the field of computables is closed anyway, so it really doesn't matter if you don't want to believe the reals exist.

Yes, apologies -- using the term 'construct' muddied the point as sqrt(2) is a 'constructible' number as you point out. The term 'real' is what's at issue here and I am arguing for a distinction between a 'map-like' real and a 'territory-like' real, the latter of which has some sort of spatiotemporal grounding.

> There are some numbers that are not, and perhaps these can truly be said to not exist.

So then we have a real issue because the vast majority of the real line is composed of these uncomputable numbers which you've suggested don't exist.

Sure, but no one uses the reals anyway for 'constructible' things. As I pointed out, the computable reals themselves form a closed field and are the things you would find when describing real life.

As for the 'problem'... I personally don't view it that way. In my opinion (and it's just that, since there's no mathematical 'truth' here), I don't believe non-computable reals exist in any meaningful way. I believe this is similar to how we talk about a 'program that can check if another one halts'. Anyone can make that statement and claim that such a thing exists, but it's not at all clear that such a thing exists. But that's a lot different than saying there are X particles in the universe, thus the number X + 1 does not exist. Because x + 1 does exist and you can write a turing machine that can compute it to any precision (or a lambda calculus function that'll give you the next church encoded representation of it, etc).

My point is two fold. Firstly that there are certainly numbers that are greater than the total number of 'stuff' in the universe. Secondly, that there are some numbers that cannot be described in any meaningful way. These can be said to not exist (my belief), but others disagree.

Sorry for not replying.

10^241: I can't find the paper from years ago I thought I would be able to re-find (if it even exists). But, Seth Lloyd has written about there being a hard limit of 10^120 operations on 10^120 quantum bits as a max computational capacity of the entire universe up to today. Whether one just multiplies the two to get 10^240 or there was another paper I can no longer find where this is explicitly done, I'm not sure. The number stuck though. Since it uses plank areas and times, it is a maximum "quantity" for a human to say physically exists according to their best scientific theories. No other physical quantity in this modest metaphysics is larger than this. So we can and must get a little more creative in how we think math relates to the world since I can talk of 10^241. It represents the universe at max capacity since inception to today, what could be greater. (If you want to get really pedantic, we could say well a person thinking of 10^241 is physical too, and science is probably somewhat wrong, ergo 10^241 is physical as is any mathematical object we can think of. What about infinity? Does it take infinity of something to think of infinity? It couldn't as there aren't infinite physical anything at our disposal up to now according to science. So if it doesn't take infinite resources to think of infinity, where is infinity in the physical. We can't just say in our thoughts. This seems hard to argue).

Regarding Aristotle, is 10^241 in the physical world? How? I personally don't want to have to argue that side. Plato's division between physical and abstract (math) say, allows me to better understand math as its own thing, and leave the mystery to how it relates to the physical world as less of a scientific question a la Aristotle.

For something like Aristotle's potential infinities existing, I'm not very knowledgeable on how he envisioned they actually do exist. Maybe it can explain how to find vast quantities in the physical. Maybe. But then is 10^241 only potential?

Funnily enough I am not a platonist about math, nor however do I believe in psychologism which I would attach to Aristotle. I'm more of a fictionalist by default but always trying to challenge this. Plato I see as setting math free from physical conceptions, which is why I gravitate toward him vs. Aristotle.

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