I was reading about Ultrafinitism and the denial of existence of $\lfloor e^{e^{e^{79}}} \rfloor$ by ultrafinitists.

I am wondering if they were to deny the existence of $\lfloor e^{e^{e^{79}}} \rfloor$ shouldn't they actually deny the very existence of $e$ in the first place, let alone forming $e^{e^{e^{79}}}$. Since $e$ in itself is defined/obtained as a limit, if the ultrafinitists were to deny the existence of large numbers then certainly the concept of limit doesn't exist for them. Am I right?

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  • That might just be a blunder in representing ultrafinitism's views on Wikipedia's part. – anon Aug 06 '11 at 00:16
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    What does it mean that a number acutally exists? – Listing Aug 06 '11 at 00:20
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    @Listing: I've always taken the existence of mathematical objects to mean that it's logically possible for states of affairs to exemplify their structure (though not necessarily in our universe). The idea of a "number" though has a lot of distinct meanings attached to it. At any rate, sounds like a philosophical project more than anything else. – anon Aug 06 '11 at 00:36
  • @Sivaram: Although the limit may not exist, there is certainly a finite-length algorithm for calculating $e$ to arbitrarily high (but finite) precision. So even finitists might believe that $e$ exists, in some suitable sense. I'm not sure about ultrafinitists though. – Zhen Lin Aug 06 '11 at 01:02
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    Did you see [this MO-thread](http://mathoverflow.net/questions/44208)? There may be interesting thoughts and pointers to follow (I read it a long time ago, so I may misremember). – t.b. Aug 06 '11 at 01:14
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    [A related question](http://math.stackexchange.com/questions/13054). – J. M. ain't a mathematician Aug 07 '11 at 05:44

1 Answers1


Note that ultrafinitism is not a single philosophy, it has various flavors and often these are not defined rigorously. Most are ideas and criticisms of the state of affairs in the classical view of mathematics. The term ultrafinitism is used to refer to several of these opinions which are concerned about feasibility of mathematical objects.

Classical real numbers (a la Dedekind or a la Cauchy) are already not finite objects, so if you want to take the classical view of real numbers as infinite objects then as a finitist you shouldn't accept $e$.

However there is a way around this. One can view $e$ as referring to a particular algorithm computing approximation to $e$. Then one can consider it as a legitimate finitist object.

If you followed the explaination above, the situation for ultrafinitism can be similar. Since the algorithm for computing approximations to $e$ is quite short and efficient, an ultrafinitist can accept its existence. Then the problem is not with $e$ but the exponentiation function that cannot be efficiently computed. In other words, there is no efficient algorithm to compute the bits of exponentiation of efficiently given real numbers. The floor of exponentiation of a given real number is a finite object, so a finitism can consider it as an actual mathematical object, however there doesn't seem to be any feasible way of obtaining its bits efficiently.

I would say there are two issues here about $\lfloor e^{e^{e^{79}}} \rfloor$ because it is not given explicitly:

  • the size of the representation of a mathematical object, and
  • the efficiency of obtaining essential information from that representation (in this case the bits of the number).

What we care about is if the object can be given explicitly. Here it is given implicitly using algorithms. If we have both of these conditions then we can consider that as a legitimate mathematical object from ultrafinitist perspective as it can be turned into an explicitly given object of feasible size. Otherwise it is not clear if it is a legitimate feasible object.

If we want to explain it more, we can think of it in the following way:

efficient algorithms of feasible size can be used to represent feasible objects

since such implicitly given objects can be turned into explicitly given objects of feasible size and this can be done using feasible amount of resources. You can think of "explicitly given" as objects given in their normal form, e.g. for natural numbers it means numbers represented like $SSSSS0$ (though decimal would also be fine at least according to some ultrafinitist views like Nelson's Predicative Arithmetic). However exponentiation is not acceptable, other than our intuition about its unfeasibility there are other reasons to regard it as so, for example you read Nelson's Predicative Arithmetic for his arguments.

(I think another argument from classical perspective can be based on non-uniqueness of definition of the exponentiation in non-standard models of arithmetic, and one can find more arguments why exponentiation should not be considered a feasible operation.)

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    I always find this argument unconvincing. There are lots of different kinds of data structures; algorithms, symbolic expressions, et cetera. $\lfloor e^{e^{e^{79}}} \rfloor$ is a rather *small* object overall. The fact that we cannot (whether feasibly or theoretically) obtain a decimal constant equal to the units digit of $\lfloor e^{e^{e^{79}}} \rfloor$ is certainly an interesting fact one would like to know about. But trying to deny the existence of this object seems silly -- IMO it's more sensationalism than substance. –  Feb 08 '13 at 06:18
  • @Hurkyl, First, let me say that I am not trying to convert you to Ultrafinitism. :) I just tried to give *an* explanation for what Wikipedia says based on what I know about it. There are various kinds of Ultrafinitism and I am just expressing one perspective. – Kaveh Feb 08 '13 at 06:40
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    Second, let me point out that mathematical objects like natural numbers have standard *definitions* which we call normal forms of them. We can later define new abbreviation for them, but that is not their definition. And obviously it is expected that we can turn these abbreviation into normal forms. Up to this point there isn't a difference, at least with constructive view of mathematics. What is different with Ultrafinitism is that an Ultrafinitist will consider such abbreviations acceptable only if they can *really* turned into their normal form *in practice*, not just in principle. – Kaveh Feb 08 '13 at 06:45
  • The importance of normal forms comes from the constructive tradition of mathematics and is not something specific to Ultrafinitism. As you probably know there is a huge literature discussing normal forms of objects. I see Ultrafinitistic ideas as another step in the direction of constructivist perspective of mathematics, if someone doesn't find the constructivist ideas serious then I think it is very unlikely for that person to consider Ultrafinitistic ideas as serious. Note that historically constructivism almost died because of similar attitudes of mainstream mathematicians – Kaveh Feb 08 '13 at 06:49
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    and only got revived after computation came to play an important role. One can expect that as complexity theory and efficiency of computation become more and more important, the Ultrafinitist ideas will be taken more seriously. ps: I don't think anyone denies the existence of this string of symbols, the issue is about whether it really corresponds to a natural number or not. – Kaveh Feb 08 '13 at 06:59
  • But what parts are serious? Are we just quibbling over what words to use? According to ultrafinitism, am I allowed to do [computable analysis](http://en.wikipedia.org/wiki/Computable_analysis) as long as I don't have the *audacity* to call the objects of study "real numbers"? I'll never be able to compute the middle digits of $2^{64}!$. Am I allowed to use it so long as I don't have temerity to call it a "natural number"? And if I'm not allowed to use $2^{64}!$, how does that reconcile with the fact I might be interested in the notion of invertible functions of type `uint64_t f(uint64_t)`? –  Feb 09 '13 at 07:33
  • @Hurkyl, I think the ideas are serious as I understand them. A mathematical object like natural numbers have a definition (say unary definition obtained from 0 and succesor function), if we want to call other things as a abbreviation for them, say binary strings, we have to give a map from them to the definition. There is nothing unusual here as I see it. The only different is what kind of maps we are going to accept. In constructive mathematics the maps need to be constructive, in computable mathematics they have to be computable, in finitism they have to be primitive recursive, – Kaveh Feb 09 '13 at 07:41
  • I *have* seen finitists argue that it is impossible to reason with and about $\mathbf{N}$. And ultrafinitists tell me there's a largest natural number. These claims seem silly. I've seen people suggest that, once you strip away all of the mysticism and sensationalism, constructivism is really just the theory of computation, and ultrafinitism is just focusing on the lower complexity classes. Is there any actual mathematical content in these philosophies beyond a desire to study certain disciplines that *already exist* in traditional mathematical form? –  Feb 09 '13 at 07:43
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    and in ultrafinitism they have to be efficiently computable (e.g. polynomial-time computable). The objection you are making can be also made about constructive analysis (I can explain it but I think it would be too long for a comment, if you are interested ask a question about how real numbers are treated in constructive mathematics and I will explain some perspectives)... – Kaveh Feb 09 '13 at 07:43
  • http://chat.stackexchange.com/rooms/7444/discussion-between-hurkyl-and-kaveh – Kaveh Feb 09 '13 at 08:02