## The Largest Number We Will Ever Need

• 7.6k
True, numbers are infinite. The simplest, the bedrock as it were, of infinities is the natural numbers {1, 2, 3,...}

Now consider the fact that in a universe that's finite there's gotta be a number that is the upper limit of a counting processes that yields the largest number possible/required to describe this universe - this number will probably be the permutation of elementary particles of that universe.

For example in a universe with 2 elementary particles (X, Y) with the constraint that the permutations allowed involve only twos/pairs or less, the largest number needed to describe such a universe would be 4 (X, Y, XY, YX).

We can call such a number Nmax.

Question: What's the Nmax for our universe?
• 2.4k
in a universe that's finite

Your supposition may not be valid. But entertaining idea.
• 7.6k
Your supposition may not be valid. But entertaining idea.

Yeah.

Nevertheless, I wanna know, how shall I put it?, the largest finite number that would be required for science. Clearly we don't know how to do math with infinity and hence our inability to wrap our heads around phenomena such as black holes (re Michio Kaku). Ergo, we need to keep numbers finite in our science equations. For instance, in the equations for black holes, why not substitute a very large number (Nmax) for infinity wherever, whenever it appears and then see what happens?
• 2.1k
Just don’t set it too low, in case we need one more.
• 7.6k
Just don’t set it too low, in case we need one more.Luke

Good call!
• 7.6k
$\frac{1}{\infty}$

If we substitute $\infty$ with 1000, we get 0.001 which to an accuracy to the hundredths is 0.00 or 0.

We go from undefined to defined (with caveats). Isn't that better than just staring blankly at a page with $\infty$ in an equation?
• 7k
No. It's better to stare blankly than to dumb down mathematics to what can fit in our small pointy heads. You'll be banning irrational numbers next, if you haven't already.
• 7.6k
No. It's better to stare blankly than to dumb down mathematics to what can fit in our small pointy heads. You'll be banning irrational numbers next, if you haven't already.

I was proposing a test-drive, you know, just to find out what happens. In all likelihood there's a number that would make us go "Yeah, this is it! It all makes sense now!" and that number is probably going to be unimaginably large but finite.

Note: We can't do (normal) math with $\infty$. That's what I'm looking to find a workaround for .$\infty$, as per some well-known scientists, pops up in physics equations more often than we would've liked.
• 99
The process of counting itself depends on sortals, which can be arbitrarily stipulated per our intension. It's also unclear if an upwards finite universe (mereologically speaking) is downwards finite: that is, if there are atoms (not as in atoms with electrons, but as in the simplest, indivisible unit with no parts.)

In other words, it is possible to count infinitely in an otherwise finite universe just in case we stipulate sortals whose extension is cardinally infinite.

With these suppositions in mind, it's important to note exactly the kind of environments that can possibly satisfy this finite Nₘ project:

• The magnitude of any value in this environment, spatial, temporal or otherwise, is a finite value
• There is no mereological descent or gunk: there are atoms
• There is no mereological ascent or junk: there are only finitary unrestricted sums
• Only joint-carving sortals count: that is, we can't posit sortals with arbitrarily infinite disjunctions to count infinite
• Nₘ is the cardinal number corresponding to the cardinality of the largest extension with regards to some joint-carving sortal
• Nₘ is finite

Question: What's the Nmax for our universe?

Unfortunately, our actual universe does not satisfy these conditions. But on the plus side, there are infinitely many possible universes which satisfy arbitrarily many Nₘ values.
• 7.6k

What about the observable universe? I read somewhere that the observable universe contains roughly 1080 atoms. That should be a good place to start at least when it comes to matter, oui?

By the way thanks for the detailed analysis of my query, much obliged!
• 99
I read somewhere that the observable universe contains roughly 10^80 atoms. That should be a good place to start at least when it comes to matter, oui?

Alright. Some extra suppositions, quarks are the simplest form of matter and are indivisible in principle. This is not currently known, of course, we have no idea if we'll discover something more basic in the future or if they really are fundamental. But for hypothetical's sake, we'll say so be it.

There are 22 quarks in an atom. So from simple multiplication N is 2.2e81 iff the only objects that exist are these quarks. Let's call this for now Nᵩ (and conceptually separate it from Nₘ).

But what about things that the quarks make up, like apples? Should we count some trillion quarks that make up the apple, and also count the apple? If we can overcount like this, Nₘ will be massively larger than Nᵩ. It can also be arbitrarily modified to be as large as we want in terms of approaching infinity (although it will never itself be infinite unless we deliberately define infinite objects).

This is because "apple" is not a joint-carving concept recognized by the universe but something imposed by us in our language; only the quarks that make the apple are recognized by the universe. Thankfully, one of our previous criteria already makes this qualification, so we can't "cheat" this number up using our massive array of sortals we can construct. That said, this notion of structure to the universe is itself controversial in the domains of metaphysics and the philosophy of science, but there simply is no Nₘ if no concepts are joint-carving and purely reflective of our language because we defined Nₘ in that structural way earlier (although this might not be true to your understanding of it in a literal sense, it's true to it in a spiritual sense)

There is also another thesis in mereology, "composition as identity," that holds that some whole and exactly the parts that make it up are identical. An apple and its quarks are identical, so it makes no sense to see "apples" as a problem because we won't be overcounting all the parts that make it up and the apple, as they're precisely the same thing (per this thesis). However, this itself is as controversial as some may find it intuitive (the controversy is to no surprise: this topic in general has very little consensus). One of the simple arguments against this view is that a whole is one and its parts are many, so this is at least one property true of the whole that's not true of the parts, so they can't be quite identical.

Well, the quarks also constitute everything else inasmuch as they compose everything else, so similarly as there is "composition is identity" there is the similar thesis of "constitution is identity." This runs into similar issues: consider a lump of clay and the statue it constitutes. We can intuitively say they are one thing, so this thesis has some intuitive force. But if somebody came to deform the statue completely, we will be no longer compelled to think the statue survived (since it is defined by its particular shape): however, we're fully inclined to think the lump of clay had survived, even if it now instantiates a distinct form. So this is another basic argument against the mirror thesis to say that they can't be quite identical. To connect this back to the apple and the quarks: the quarks will survive if someone ate the apple, even if they may now instantiate a different form altogether. But we'd no longer say the apple exists, perhaps some human waste and other chemicals now exist, so clearly the things that the quarks constitute and compose are not quite identical to them.

This motivates the idea that including our natural language concepts, referred to by our ordinary nouns and proper names, forces us to overcount. We can restrict the overcounting to be finite, but since we can make our language whatever we want it to be, we can still count to any arbitrary finite number, giving us no clear answer. While we can simply discard those, per our earlier criterion, it goes back to something else you said:

Now consider the fact that in a universe that's finite there's gotta be a number that is the upper limit of a counting processes that yields the largest number possible/required to describe this universe

When it comes to us describing the universe, we usually don't care about joint-carving the structural properties of the universe itself whatsoever. When we ask "how many apples are on the table?", quite literally no one means "how many independent apple-forming quarks are on the table-forming quarks?", returning some massive number. We just mean our own language, instrumental to our utility, regardless of whether it reflects the universe's structural properties.

However, we still think we're meaningfully describing the universe: at least in the way it appears to us.

In omission of our ordinary concepts, we remove the "cheating" problem of overcounting to any finite number, but this also removes otherwise true descriptions of reality that exceed Nᵩ (it's not exaggerating to say, per the post's title, that we need these descriptions, sometimes for our own survival). But, when we include those ordinary language descriptions, Nₘ itself loses its intended meaning and becomes whatever we suppose it to be per the intensions of our non joint-carving concepts. Either we can't describe everything we need to describe (in a practical way) or there is no principled Nₘ, its value is only a function of our choice! We're at a dilemma here :(

By the way thanks for the detailed analysis of my query, much obliged!

Cheers,
This was a fun write!
• 7.6k
Yup, the number Nmax would differ with what one chooses to count as 1.

That said, I was hoping to find a number such that

1. No calculation ever would exceed that number

2. if in an equation with $\infty$ in it, replacing $\infty$ with the number (Nmax) allows us to, at the very least, approximate the state of affairs, mathematically.

Physicists tend to throw their hands up in the air with disgust mixed with utter frustration when they see $\infty$ when number crunching. My post is an attempt to, as you rightly pointed out, cheat the system if possible.

• 10.4k
There are 22 quarks in an atom. So from simple multiplication N is 2.2e81 iff the only objects that exist are these quarks. Let's call this for now Nᵩ (and conceptually separate it from Nₘ).Kuro

On what principles would you decide how to count all the dark matter?
• 99
On what principles would you decide how to count all the dark matter?

Like I said, most of this is really speculative. A candidate particle is the d-star hexaquark d*(2380) which is hypothesized to account for the universe's dark matter. This particle is composed of six quarks.

Then again, there's also the debate about ontic structural realism and other types of scientific realism versus instrumentalism: whether our best physical categories really carve the universe or whether they're just useful to us. I'm obviously assuming the former is true so that this question is in principle answerable.
• 999
As long as humans exist there is no Nmax. Someone will always pitch up to propose the last number suggested +1. And without humans there's nobody round to ask or answer the question.
• 3.9k

Question: What's the Nmax for our universe?

Why? Are you planning on investing in real estate?
• 2.4k
That said, I was hoping to find a number such that

1. No calculation ever would exceed that number

Why is it I think you are not serious? :smile:
• 999
You can be serious and playful at the same time in philosophy. That's part of the fun.
• 2.4k
1. No calculation ever would exceed that number

Sure, just fix whatever large number you wish and round off to that number. But you might make a mess of computations that follow. In physics renormalizations work in various settings.

Today, the point of view has shifted: on the basis of the breakthrough renormalization group insights of Nikolay Bogolyubov and Kenneth Wilson, the focus is on variation of physical quantities across contiguous scales, while distant scales are related to each other through "effective" descriptions. All scales are linked in a broadly systematic way, and the actual physics pertinent to each is extracted with the suitable specific computational techniques appropriate for each. Wilson clarified which variables of a system are crucial and which are redundant.
Wiki
• 7.6k
Why is it I think you are not serious? :smile:

:blush: I'm serious but looks like my idea is ridiculous.

You can be serious and playful at the same time in philosophy. That's part of the fun.

Muchas gracia señor! You took the sting out of jgill's remark.

Sure, just fix whatever large number you wish and round off to that number.

Well, I was hoping that a seasoned mathematician like yourself could bring some precision to Nmax. For the last coupla years I've been thinking of very large numbers and the way I do it is pick a number at random and raise it to a power that's large and also random e.g. $23^{2017^{3138934}}$. Much to my amazement I found a youtube video on the topic of thinking of a number which no human has ever thought of on the channel Numberphile. It seems that if one is systematic and methodical one can do better than just guesswork.
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There's an infinite number of numbers which no human has ever thought of. What's the point in trying to name a random one of these? Here's one for you though, which might be worthwhile. Try naming pi to its final decimal place. That's a meaningful number which no one has ever thought of.
• 7.6k
There's an infinite number of numbers which no human has ever thought of. What's the point in trying to name a random one of these? Here's one for you though, which might be worthwhile. Try naming pi to its final decimal place. That's a meaningful number which no one has ever thought of.

Most interesting. — Ms. Marple
• 1.7k
describe this universe

'Describe the universe' is a notion not defined by you.

Clearly we don't know how to do math with infinity

Clearly that is false. Infinite sets are basic for calculus.

No calculation ever would exceed that number

Let that number be M. Then M+1. Poof.

Physicists tend to throw their hands up in the air with disgust mixed with utter frustration when they see ∞∞ when number crunching.

pi to its final decimal place

There is no last decimal place of pi.
• 7.6k

Good points, worth pondering upon.

Let's back up a little for my sake.

Is there a finite number (Nmax) such that no calculations ever in physics will exceed that number?
• 2.4k
Is there a finite number (Nmax) such that no calculations ever in physics will exceed that number?

From special relativity, the Lorentz factor is unbounded as v approaches c.

$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$
• 7.6k

Most interesting. — Ms. Marple

Infinity is then some kinda test for impossibility, at least in this case. We need infinity to demarcate the boundary of what's doable and what's not.

In a sense then $\infty$ in science has a job description similar to contradictions - separates the possible from the impossible.

Is infinity a contradiction? It does lead to some rather odd conclusions: a part is equal to the whole and all that. No wonder many mathematicians (recall Kronecker's vitriol against Cantor) were dead against it.
• 1.7k
a part is equal to the whole

No, set theory does not say that there is a proper subset of a set such that the proper subset is the set. Set theory does say that there are sets such that there is a 1-1 correspondence between a proper subset of the set and the set.

This is another example of you running your mouth off on this technical subject of which you know nothing because you would rather just make stuff up about it rather than reading a textbook to properly understand it.
• 1.7k
Is there a finite number (Nmax) such that no calculations ever in physics will exceed that number?

I know so little about physics or cosmology that I can't answer that.
• 7.6k
No, set theory does not say that there is a proper subset of a set such that the proper subset is the set. Set theory does say that there are sets such that there is a 1-1 correspondence between a proper subset of the set and the set.

This is another example of you running your mouth off on this technical subject of which you know nothing because you would rather just make stuff up about it rather than reading a textbook to properly understand it.

:lol:

I know so little about physics or cosmology that I can't answer that.

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You should not be disappointed. You should be encouraged. I gave you an outstanding example: when one does not have sufficient knowledge then one should defer from making wild claims.
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when one does not have sufficient knowledge then one should defer from making wild claims.

The problem isn't me, mon ami - if a particular topic is conducive to wild claims then something's wrong with the topic. I have no truck with people who make claims that are far out.
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