I came across the ultra-finitism, the idea that there is a "largest number". Even most ultra-finitists admit that the "largest number" cannot be exactly defined. Therefore my question :

Did some ultra-finist make any suggestion of at least an approximate value that should indicate the "largest number" ?

If not, why is ultrafinitism accepted as an alternative way to do mathematics ?

I can understand that some mathematicians do not like the concept of infinity. But I cannot understand that the idea to reject FINITE numbers that can be easily defined mathematically , as Graham's number , is mathematically legitimate in any sense.

I know that there were already some questions about ultrafinitism, but I did not find anywhere a suggestion which numbers are considered to be "too large".

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    $\begingroup$ This might be of interest: mathoverflow.net/questions/44208/… $\endgroup$ – wythagoras Jan 9 '17 at 18:23
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    $\begingroup$ Can be quite hard to define precisely the "largest" finite number, meaning the "largest conceivable" or "larger manageable" number... Assumig that we define it a a certain specified number $N$, it immediately poses the problem of $N+1$ being equally conceivable or manageable. $\endgroup$ – Mauro ALLEGRANZA Jan 15 '17 at 14:11
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    $\begingroup$ Can't understand finitists unfortunately. $\endgroup$ – Simply Beautiful Art Dec 8 '17 at 1:48

Fix any representational scheme for natural numbers. Then there is simply some number for which the most compact representation in this representational scheme just barely fits in the physical universe with any embodiment that our current understanding of physics deems doable. The usual retort, "well, what about that number plus one?" doesn't work since the expression $\ulcorner\text{that number}\urcorner+1$, even assuming it is a valid expression in the fixed representational scheme, takes slightly more "space" and thus doesn't fit in the physical universe. The next retort is usually something like "well, doesn't 'the largest number that fits in the universe' provide a short description of that number?" Again, such an expression is unlikely to be an expression in the fixed representational scheme, and a representational scheme that allows (some formalization of) that expression is likely deficient in many other desiderata of a numeric representational scheme, e.g. decidable equality.

Now let's fix the usual representational scheme we use: positional notation with Hindu-Arabic numerals. Assuming we can represent a digit in the state of an atom (we can probably do better than that, but probably not several orders of magnitude better than that), then the largest number representable is something like $10^{10^{80}}$. Of course, advances in our understanding of physics and in technology could significantly change this though probably not make it $10^{10^{10^{10^{80}}}}$. Restricting ourselves to what we can actually do is a common theme in ultrafinitist thought, and currently known physical limits are often used as a rough upper bound. This means those upper bounds can changes as our abilities and understanding improve. Of course, the practical limits are far lower but also more sensitive to our technology. Computers have rendered quite feasible many calculations that would have been effectively impossible before their existence.

What about more flexible schemes, say arithmetic expressions involving addition, multiplication, and exponentiation with all literal numerals represented with positional notation? Well, the "largest representable number" grows enormously to something like $10\uparrow\uparrow 10^{80}$ but a different problem appears. Positional notation has the benefit that the size of the representation grows monotonically with the magnitude of the number. This is emphatically not the case with this representation involving exponentiation. $10^{10^{10^{10^{80}}}}$ and $10^{10^{10^{10^{79}}}}$ are easily written down in this representational scheme, but the vast majority of numbers between those two numbers cannot be represented in the physical universe with this scheme. This representational scheme makes the natural numbers have a fractal structure where there are islands of relative simplicity in oceans of overwhelmingly complex numbers. So there is still a "largest number" but now below that number the natural numbers look like Swiss cheese with large gaps of numbers missing.

Arguably the "right" way to think not in terms of a "largest number" but in terms of efficiency. Classically, something is "true" if rejecting it is absurd. Constructively, something is "true" if we can construct a demonstration of it. Ultrafinistically, something is "true" if we can efficiently construct an efficient demonstration of it. (Ultrafinitists don't like impractically long proofs either.) I was going to talk about some relatively recent results in implicit computational complexity that might serve as a suitable logic of these ideas, but Neel Krishnaswami's answer to the question referenced by wythagoras says everything I was going to say and then some. The key tidbit here is in Light Affine Set Theory (LAST), multiplication is total but exponentiation is not, and the Peano numerals are not isomorphic to the positional numerals.

A theme you've probably picked up by now is that to an ultrafinitist representation really matters. The Peano numerals and the positional numerals are not just different ways of representing the natural numbers, but different things entirely. They support different operations, and while we can embed the former into the latter, we can't the other way around. From the perspective of LAST, the embedding of position numerals into Peano numerals is exponential time and thus not provably total and thus not a function. In terms of my earlier statements, the class of natural numbers that are representable as Peano numerals in the physical universe is a strictly smaller class than the class of natural numbers representable in the physical universe in positional notation. A similar splitting of concepts occurs when we move from classical to constructive mathematics, but I'm fairly confident the situation moving to ultrafinitistic mathematics would be far more extreme.


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