# How should one interpret modern mathematics if one doesn't believe in infinity?

I am an ultrafinitist. http://en.wikipedia.org/wiki/Ultrafinitism

I don't believe there is a such thing as infinity. To me, it is obvious that there has to be a largest number; I just don't know what it is.

It is an axiom of modern mathematics that infinity exists, so modern mathematics contradicts what I believe; however, I think it would be a major mistake for me to throw out all of modern mathematics only because of my silly hangup. After all, modern mathematics works.

So how should I interpret modern mathematics?

• The simple answer would be "take that which does not depend on the existence of infinity". I assume you're looking for an answer more sophisticated than that, but then it may be useful to refine your question.
– user2953
Jun 5 '15 at 17:12
• "To me, it is obvious that there has to be a largest number; I just don't know what it is."- Why do you think it is obvious? Jun 5 '15 at 17:15
• The "belief" in infinity that is so "common" to modern mathematics is simply the way to address the following paradox : "What happens when we take the largest number and add one to it ?" Jun 5 '15 at 17:51
• Zero does not work, because this choice contradicts the other fundamental assumption that zero is not the successor of any number. Jun 5 '15 at 18:16
• The "human mind" has no overflow : it can consistenly think of the successor of any number, however great. Jun 5 '15 at 18:17

Disbelieving in infinity is going to cause you problems only if you are a mathematical realist, meaning that you believe that a number like 5 has some independent ontological existence that infinity lacks. In the case that you think that both 5 and infinity are just useful concepts then the problem disappears.

There are plenty of concepts we find useful in life and in science without assigning to them any larger or deeper reality. For instance, in his book QED, Richard Feynman, the great theoretical physicist, asks us to picture light as little rotating arrows. He is in no way a realist about those arrows. He does not think that light is a little rotating arrow at any magnification, or under examination by any type of instrument. But he does find it useful to conceptualize light as rotating arrows, because it makes certain difficult theoretical concepts and calculations easier to visualize. Even in the case that you want to be a realist about 5, you could still maintain that infinity is a convenient fiction, helpful for calculations of some types. In that case, however, you might be asked to explain what makes one of those numbers "realer" than the other.

It's worth noting, however (as pointed out in the comments), that actual ultrafinism goes beyond merely disbelieving in infinity. It includes a commitment to not dealing at all (even as useful fictions) with numbers that can't be reasonably constructed. Given that, if you are in fact an ultrafinist, you may need to choose between that commitment and significant portions of modern mathematics.

• Douglas Hofstadter has a great intuition about this matter. He proposes in GEB, that we establish "isomorphisms" between the real world and the meaning we assign to symbols. We can't prove these isomorphisms to exist directly, but we can accumulate evidence for it. In the case of the concept of small integers it is very obvious, so we believe that the first rules of arithmetic with small numbers correspond to things in reality. From this we induce that it also works for (infinitely) large numbers. It looks like induction works for all practical purposes, but we can never be entirely sure. Jun 28 '15 at 13:59

It seems to me there is a fundamental contradiction between two parts of your question.

First you say: "I am an ultrafinitist".

Then you ask how you should interpret modern mathematics.

But ultrafinitism is an interpretation of mathematics. So either you subscribe to that interpretation, in which case you surely have no need to ask what it is, or you do not subscribe to it, in which case you are not an ultrafinitist.

• You are just playing around with definitions. Jun 5 '15 at 22:04
• If you're using words in a way that's incompatible with their standard definitions, then you need to tell us what your definitions are. If you're using words in a way that's compatible with their standard definitions, then you do need to live with the consequences of those definitions. Jun 5 '15 at 22:23

If you read "Modern Mathematics" as "Zermelo-Fraenkel Set Theory possibly with some extensions and additional definitions" then certainly an ultrafinitist will need to make sense of the Axiom of Infinity. Perhaps you might simply want to say that there is no such set as the axiom describes; if so, then your proposal will have consequences for e.g. continuous functions in analysis. This is a position that is defended by some ultrafinitists such as Doron Zeilberger who argues for instance that all true continuous analysis reduces to discrete analysis.

Another position would be to try to rehabilitate the axiom of infinity somehow - that the set described by the axiom exists but is not truly infinite; perhaps for instance we think the "infinite set" as a fallout of some forcing extension of the actual finite set-theoretic universe. (Although forcing is probably untenable to the ultrafinitist, alternative reinterpretations have been suggested, for example Yessenin-Volpin use of inner models)

One more alternative would be to simply forgo the Set Theoretic foundations approach altogether in favour of frameworks more friendly to combinatorial methods. An example for this (though not necessarily itself to be read as a finitistic programme) is the Homotopy Type Theory programme. HTT takes a more abstract algebraic view of mathematical foundations, which would ideally aim to avoid any kind of set-theoretic commitment talk independently of the particular structures and algebras it might be studying. Avoiding an explicit commitment to the infinitistic practices of set theory might perhaps be more in tune with what an ultrafinitistic view of mathematics is at root concerned about, without necessarily putting the demand on them to actually explain how to mitigate the loss of "transfinite numbers" as such.

It is an axiom of modern mathematics that infinity exists, so modern mathematics contradicts what I believe; however, I think it would be a major mistake for me to throw out all of modern mathematics only because of my silly hangup. After all, modern mathematics works.

If your premises lead, logically, to a contradiction, then at least one of your premises is wrong. I suggest in this case ultrafinitism is wrong, not modern mathematics.

I don't believe there is a such thing as infinity. To me, it is obvious that there has to be a largest number; I just don't know what it is.

First, there is a contradiction here.

Lets call the largest number "N". Then the half of N is N/2 - which is obviously less than N, and so it is not the largest number. Then there is a number that is N/2 + 1. Such number is also less than N, unless N is two (and, since we have five fingers in each hand, and five is larger than 2, it seems quite easy to assume that N is larger than 2). But now we have a problem. For all numbers equal or smaller than N/2, it is true to say that all numbers can be multiplied by two: (1 X 2 = 2, 2 X 2 = 4, 3 x 2 = 6... (N/2) X 2 = N). But N/2 + 1 cannot be multiplied by two, for that would be N + 2, which is larger than N, and consequently is not a number, for N is, by definition, the largest number. And so, some numbers can be multiplied by 2, and others cannot. Or that they can, but then the result is not a number (what are they, then?). This will make any mathematical system inconsistent.

Second... I don't believe in circles. And while you cannot prove that there is a largest number, I can even prove that circles do not exist. If matter is made of atoms, then any circle that has x atoms in its radius must have a circumference of πx atoms - which is impossible since π is irrational. So, circles don't exist.

Yet, I pretty much know what a circle is, and when and why to use such thing. It is an abstraction; as Chris Sunami puts it, it is a useful concept. You can't have modern mathematics without it; but, worse, you can't have Euclid, or even Pythagoras, without it.

And so, I think this is the problem:

Physically, it is possible that there "is" a largest number in the universe: the number of the smallest particles, or of the smallest possible length, or area, or volume, that exist in the universe, or the multiverse. But any mathematician, or even a pesky layman, can then say, "that number... plus one", "that number... times two", "that number... squared". And while such numbers are larger than the number of countable "things" in the universe, they are still numbers, for any of the commonplace arithmetic operations can be performed to/with them.

Mathematics is not a "science" in the sence that Physics, Biology, or Sociology are. It is a method, and a method that can be applied to things that exist, and also to things that do not exist.

Four unicorns are still twice as much as two unicorns.

How should one interpret modern mathematics if one doesn't believe in infinity?

Answering your question in the strict and literal sense: If one does not believe in infinity one should say farewell to mathematics. Do not change mathematics, let's change our mind :-)

But I assume that you are seriously interested in the concept of infinity in mathematics, let's say at least since the times of Cantor. Hence I recommend to study a bit Cantor's theory of transfinite numbers, e.g. reading

"Joseph W. Dauben: Georg Cantor and the Origins of Transfinite Set Theory. Scientific American, Vol. 248, No. 6 (1983)" (If you do not have access to the Journal I can send you a copy.)

A saying of Hilbert states:

"No one shall drive us from the paradise which Cantor has created for us."

• There are at least two serious mathematicians who are ultrafinitists: The late Edward Nelson, and the living Doron Zeilberger. Should they say farewell to mathematics? Secondly, what of the neo-intuitionists, constructivists, and type theorists who abandon LEM? Should they say farewell too? They're arguably winning the debate lately. ZF is barely a century old and foundations are historically contingent. What say you? Feb 20 '19 at 20:11
• @user4894 A mathematical object exists in the context of a consistent mathematical system if and only if the concept relates to other concepts of the system and does not introduce inconsistencies. The OP claims that in his opinion a largest number exists. Is the existence of a largest number free from contradictions? – Concerning your numerous other interesting questions I propose to transform each of them separately to a single statement with a precise question. Feb 20 '19 at 21:02
• `If one does not believe in infinity one should say farewell to mathematics.` what does that mean? If it means `If one does not believe we can conceptualize infinity one should say farewell to mathematics.` I would agree. But I don't think you need ontological existence of infinity for modern mathematics. Feb 21 '19 at 1:30
• @jbyseribpngf What do you mean by 'ontological' existence? - Concerning the need for infinity for modern mathematics: Take projective space, the basis of algebraic geometry. Projective n-dimensional space arises from n-dimensional affine space by adding a (n-1)-dimensional hyperplane at infinity. E.g., the 1-dimensional projective space arises from the usual 1-dimensional line by adding one point at infinity. Feb 21 '19 at 6:19

You say you want a way to interpret mathematics. This is a universal cry: we all want to interpret mathematics also, whether we are ultrafinitists or not. Mathematics itself is sufficiently abstract that it must be applied to be useful.

Fortunately for you, this is a good thing. It's abstract enough that you are free to interpret it as nothing more than "a set of symbols." At the most fundamental level, that qualifies as an interpretation of mathematics.

I am assuming your goal is to actually use that mathematics. To be precise in the wording, you seek to transform a mathematical phrasing (such as "1 + 1 = 2") into a form which allows you to act upon the world around you (You ask for two coins. I give you one coin, then I give you one coin. We are even). This is where the ultrafinitism comes into play. You should have no trouble with the definition of natural numbers, N. People are free to use any wording they please, even silly ones. What you should have trouble with is the application of those wordings in ways which affect your daily life. If I can tell you that you can walk on water, using the cardinality of the natural numbers as part of my mathematical proof, you should have some skepticism of the usefulness of my claim.

Very very little of these applications actually cope with infinity directly. Usually they use infinity as part of a proof. Proofs are associated with truth and falsehood. A proof proves that a theory is true. If you have a valid proof, it matters not how absurd the result is, because it's proven to be true.

Accordingly, you can look at modern mathematics. Much of it actually doesn't need infinities. However, many proofs of statements involving finite numbers will rely on infinities, such as those found in mathematical induction. These are the tricky ones.

Going back to what you are really trying to do, you are converting your "reality" into a mathematical picture, proving something about it, and then applying the result back to reality. For those who believe in infinity, the proof is sufficient. For you, you may have to use some intuition because someone is telling you something that is probably useful, but is unprovable. If the stakes are low, it may be effective to humor them, and use their shortcut involving impossible numbers. If the stakes are high, you should probably try to find a way to prove the statement without infinities.

Failing that, you are free to translate their phrasing with a potential inconsistency. Every time they rely on an infinity, you are free to argue "it is possible (though not necessarily provable) that there exists some arbitrarily large number for which this theorem is inconsistent."

There is a related issue that comes up in mainstream mathematics, known as the Axiom of Choice. The Axiom of Choice is added to Zermelo-Frankel set theory (a set theory involving infinities, I know) to state that "given a set of sets, you can construct a new set by pulling one element out of each of the original set of sets." In finite land it makes sense: "if I have 10 bags full of candy, I can create a new bag of candy by taking one piece of candy from each bag."

In infinity land, it gets wonky. When you have sets of infinite elements you can do strange things like start with a sphere, dice it up into 5 sections, rotate each of those 5 sections and produce 2 full spheres of the same dimensions. The trick is that you dice it up into disjoint infinite sets of points. This is so unruly that many choose not to accept Zermelo-Frankel with the Axiom of Choice (ZFC), and instead only use Zermelo-Frankel (ZF).

However, even those who refuse to accept ZFC watch what proofs go on in that field. Many proofs which are now accepted using only ZF were first proven using ZFC and later improved to be independent of the axiom of choice. Thus those developing ZF theories can treat ZFC as a source of inspiration, suggesting where they should look next for a proof.

If you are a true physicalist, then in practice, given the limits of time and process there is a largest number that will ever be used. That does not mean it is in principle some magical kind of limit, but numbers beyond that are simply irrelevant. But who are we, now, to decide what number that will be? Why not be halfway modest and act as if it is far beyond anything we can imagine? Why not plan for a long future?

You choose that direction, you can abandon infinity, but you have to allow for continual increase, anyway.

From a 'Nonstandard Analysis" point of view, the elements of sets like the Real numbers, or the integers with infinity are not real, but are axiomatic definitions masquerading as things. (Two is the property of having distinct things but as few as possible, etc.) Countable infinity, as the number with every number you have encountered as predecessors, but no immediate predecessor, is the shorthand for encoding continual increase. It is that biggest number ever used, forever getting away from us, slipping away into the future.

Any such axiomatic definition, expressed in a pattern that can be written down, is a recognition mechanism, that can be rearranged for use as a generating mechanism.

For instance, the native Intuitionist model (a la Brower) of a real number is "a freely flowing stream of bits." Every real number is process that will always hand you the next digit of precision. The number itself is treated like a point in space, but underlying it is really an ongoing approximation.

Given the notion that any rule can be looked at as a process, all other useful applications of infinity can be re-encoded in a similar form.

So it is perfectly reasonable to think of the numerical parts of mathematics as good thinking about measurements and approximations and their ultimate limitations even when you are 'taking limits as x goes to infinity', or dividing two things both 'going to zero'.

Things like infinite groups, etc. abstract that underlying mechanism away, assuming it can be captured faithfully in an intuition and ripped away from its more concrete forms. If you are not willing to make that leap, then you can stick to geometries and finite structures, and assume the nominally infinite ones do not have any applications that will interest you.

If you do make that leap, you have moved from computation to psychology. By making assumptions that human intuitions around things like infinity or continuity have an interest of their own, and that the fascination we feel for them has some basis, you can embark on a kind of deeply psychological art, either out of interest in the psychology, or attraction to the art.

Some of the products of that art turn out to have representations in reality, that make certain kinds of other things easier to imagine. Much like other kinds of stories help us get through life. But these stories are always 'Roman a Clef', we know where the characters come from. So the representations can be unwound back into finite terms and modeled in computation when they have genuine applications.

The question is why we can get from computation to art and back to computation easier by allowing ourselves a certain level of excess in the art than we can by sticking with reality. Basically, why is the human mathematical intuition a stronger tool than its motivation, if everything it models beyond its concrete applications is really not there?

It is the same question that makes language fascinating. If the universe is basically physical and evolution is what drives most of this, then why on earth would we evolve something so much more powerful that evolution itself, (solving the same kinds of problems hundreds of times faster) and then use that to create another kind of evolution altogether (competing ideologies and cultures)?

(Reality is enough. No one needs lies. But as Nietzsche points out, we have not yet begun to even estimate their power.)

Physics in a sense is ultra-finitist in that actual infinities are usually considered to be signifiers of a failure (or aporia) in a theory; and this notion is actually of ancient provenance - Aristotle for example in his Physics argued that only potential infinity obtains in the world.

So here you are in good company.

There are two ways in which mathematics uses infinity - the infinitely small in the calculus, and infinitely large in set theory; and the normal attitude is either platonic, formal or pragmatic.

Pragmatic has already been covered by the response by Sunami; formalism in some sense arises out of pragmatism - since we care only if it works and not if the concepts therein has ontological weight then we need at bottom only be concerned with logical consistency.

Platonism, is what might be called mathematical realism where one cares about the ontology of numbers; and it appears that you do - then one needs to think in this space; and I'd suggest that there are two related possibilities here constructivism which allows a number only if it is constructable, ie I cannot just assert its existence; and intuitionism which drops the LEM (Law of the Excluded Middle).

• Interesting for the Aristotle's part. After all, how could we interpret the infinity in the sense of its potentiality??? If some day, someone found a final answer, can we say at that time "Dude, you've been calling this infinite, but I found the answer finally, brother. Now say what you want" -) -)
– user13955
Jun 6 '15 at 16:39
• @KentaroTomono: potentially infinite just means given a number n there is a larger one; an ultra-finitist would say there is a maximally large number; a set theorist will just take the 'completion' of all these numbers which is not potentially infinite but actually infinite ie it is not finite... Jun 6 '15 at 16:52
• Oh. Thank you. However, I am sorry isn't it still potential? Given a number n, potentially there would be n+1 and it goes on. How far is goes, it is still pontentially n+1. I am afraid to say then can I call it infinitely potential which wouldn't mean nothing at all?????
– user13955
Jun 6 '15 at 17:11
• In another words, we might be able to call it alternately we just don't know when, until we die, it would remain potential. Sorry for this.
– user13955
Jun 6 '15 at 17:13
• Well, there is a specific number called omega which is larger than every natural number n; however you can still carry on with omega+1 and so on; the infinitie and it's siblings not to say its sons and daughters are full of cunning subtleties... Jun 6 '15 at 17:20

I think it's eminently possible to deal with modern mathematics while believing that infinity does not exist. Technically, if the belief had to be applied in one's mathematical considerations, then it would quickly lead to self-contradictions and nonsense results. But just as a religious person who believes that everything happens according to some god's or gods' will, can go on making decisions, caring for others etc., so a person who disbelieves the existence of infinity, or for that matter multiplication, whatever, can go on doing math as if those things existed, just disbelieving that they really exist.

It's a matter of a little double-think, a context-dependent suspension of belief/disbelief in at least one part of the mind. We're good at double-thinking. Even a person who doesn't harbor any great irrational beliefs like those mentioned above, have to double-think on occasion, because it's physically impossible to always have consistent beliefs about everything. And sometimes one discovers, through the consequences of one's beliefs, that one harbors two or more mutually contradictory beliefs – double-think to the rescue!

Looking at that the other way, it's physically impossible to be sure that all one's beliefs are consistent because it would require arbitrary high processing capacity to deduce the relevant consequences where one can see directly that they clash. So, double-thinking it is, if one is not be stopped by such happenings, which could be seriously detrimental to one's continued existence. And having evolved to accommodate contradictory beliefs, we (apparently) do it quite easily.

Well I don't think modern mathematicians believe infinity exists. We do use the term infinity and introduce objects such as infinity to form the extended real numbers and when discussing mobius transformations, but one should not read too much into this. First of all, in the second case, the complex numbers have no order structure, and in the first, if one accepts the real numbers are without bound then why trouble oneself with the introduction of finitely many objects.

I understand your present point is that in fact the real numbers are not without bound (or more precisely the natural numbers are bounded above) and so are necessarily finite (because it is impossible to get to the reals without going through the natural numbers or talking about infinity - in fact, if we didn't have infinity we couldn't talk about the real numbers).

My point above is that when one is discussing infinity in a formal mathematical sense, it is because one is discussing a statement concerning an 'infinite' set. One of the axiom schemas - funnily enough, the axiom of infinity - is as follows:

The exists a set N such that the empty set E is in N, and if x is an element of N then the set whose elements are x and {x} is also an element of N.

This is purely a formal definition and in no way relies on or mentions infinity - one may think about it as saying N is infinite but from a formal standpoint it is nothing more than these symbols.

Therefore one may say that infinity derives meaning from its use in a suitable set-theoretic language rather than meaning from somehow being 'infinty' though proper.

I recommend considering the "Flat Earth" model of planar measurement even though we know the Earth is a sphere. We do not measure distance in arc, but in linear units. This works well for road trips, building developments and probably most land surveying (although I suspect that air travel measures distance on an arc). The reason we can measure linearly despite the formula for doing so actually being inaccurate, is that it's close enough to the real measurement so that it remains useful.

So to answer your question, you could interpret modern mathematics as approximately accurate to the reality you're convinced exists (this is not a criticism to your decision of the denial of infinity). Modern mathematics can be seen as "close enough" and thus there's no problem in using it as-is. You will just remain aware of its limitations and know that it may not be representing reality (again, the reality that you accept) 100% accurately.

I also agree with @Keelan's initial comment and @Cheers and hth. - Alf's exposition on "double-think".