What is the status of irrational numbers within finitism/ultrafinitism?

Question

According to constructivism, "it is necessary to find (or "construct") a mathematical object to prove that it exists". There are several formulas to calculate $\pi$, such as:

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so I take it $\pi$ exists according to constructivism.

According to finitism, which is a form of constructivism, "a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps."

Where does that leave irrational numbers, such as $\pi$? Do they simply not exist according to finitism? How does one reason about the ratio between a circle's circumference and its diameter, if one is working within a finitistic/ultrafinitistic framework?

Answer 1

Primitive recursive arithmetic (PRA), mentioned on the second Wikipedia page to which the question links, is, I believe, generally accepted as an appropriate formalization of finitist foundations. The appropriate way to deal with numbers like $\pi$ when working in this theory would be to use a primitive recursive function that produces (natural number codes for) a sequence of approximations to $\pi$ (with a specified rate of convergence). I'm not sure one could even formulate, in this context, the statement "$\pi$ exists", since the theory's quantifiers range only over natural numbers. That does not, however, prevent one from working with $\pi$ and proving appropriate formalizations (in terms of primitive recursive algorithms) of basic facts about $\pi$.

You shouldn't worry about the problematic "existence" of the ratio between a circle's circumference to diameter, since the "existence" of the circle itself seems just as problematic.

I suspect that, if people want to work in the context of PRA, they actually work in a conservative extension, where many more entities (like $\pi$) are available without making new theorems provable about the original entities (natural numbers) --- just as when most mathematicians work in the framework of ZFC set theory, they don't limit themselves to the primitive notion $\in$ but introduce lots of definitions and abbreviations, and perhaps talk about proper classes (as in NBG class theory).

Answer 2

Claudiu, the topic of ultrafinitism is quite slippery, and by no means well-defined (unlike other more conservative forms of constructivism, such as intuitionism, Bishop's constructivism, and so on), at least not yet.

I suggest that you go to FOM, and google ultrafinitism, to have an idea of what some folks think about the subject. For instance, some "ultrafinitists" are ACTUALISTS, meaning that they believe in some physical max integer number, whereas others (such as myself) are not.

I do not believe in ANY number (in the ontological sense), not just huge ones. In my perspective, which I think could be better characterized as ULTRAFORMALIST, syntax is all you have, and "numbers" are simply the by-products of playing the arithmetic game according to the arithmetical rules + basic logic.

There is, though, or so it seems to me, a common thread among ultrafinitists, namely the general mistrust for every form of infinitary reasoning, included so-called potential infinity (both at the mathematical and meta-mathemathical level).

Now, from my perspective, as Qiaochu has already observed, $\pi$ does exist, in the sense that there is a very finite object (the algorithm), to compute it, at any degree of precision you like (provided, of course, that you have the computational resources to carry out the computation). Think of a computer program: your laptop "understands" what $\pi$ is, and then it simply produces its digits. At some point it will stop, because it runs out of gas. Your formula has to be emended so that it says: IF you can compute up to some N, you can compute $\pi$ accordingly with the given precision.

You question can be generalized to a much broader one, namely what exactly is the viewpoint on mathematical objects at large from such a perspective. I think one day (in my opinion quite soon) we will have such a viewpoint fully articulated, and it will look somewhat like Brouwer's theory of species, only without giving free tickets to any type of infinitary reasoning.

ADDENDUM: even actualists, for instance Doron Zeilberger (i hope I understand him correctly), would accept your PI, only they would argue that its status is symbolic, as opposed to concrete numbers (say 5). They have no problem in manipulating symbolic numbers, they simply do not think they correspond to anything "out there".

Answer 3

Even while considering the universe to be finite, one can do mathematics symbolically as a game with a system of rules. If the game doesn't have enough pieces we just add new pieces, with new properties or allowed moves as required. All that matters is that the enlarged system is compatible with the old system; that the smaller game is a subgame of the large one; that the smaller system can be embedded in the larger one.

When does something exist ? Well if there isn't something from amongst the objects under consideration that has the properties we want then we just create new symbols and define how they relate to the old ones.

If we were a pythagorean and the only numbers that exist are rational numbers, then we wouldn't call $\sqrt 2$ a number, but if we were also finitist symbolicists then we could embed any collection of numbers into a collection that contains not only numbers but also "splodges" which is what we're going to call $\sqrt 2$. It's important to always keep in mind that $\sqrt 2$ isn't a number - it's a splodge. In this new system of arithmetic we've invented we can add numbers to splodges to get new splodges like $1+\sqrt 2$. What a fun game. Let's add some more splodges. We're bored with algebraic splodges so let's add some non-algebraic splodges like the one in your question. Of course that expression is a bit cumbersome so we'll give it the shorthand symbol $\pi$ instead.

Given a splodge $x$, it would make calculus easier if there were a splodge $x+o$ that was nearer to $x$ than any other splodge, however that isn't possible so we embed the splodges in a larger system called the hypersplodges that contains not only splodges but also vapors, and contains not only the concept nearer but also the concept "nearer". Vapors like $x+o$ are "nearer" to $x$ than any splodge could ever be, and when you're finished using them they evaporate leaving just a splodge.

We want a splodge that satisfies $x^2+1=0$ however there isn't one, so we embed the splodges in a larger system called weirdums in which we've added a piece called $i$ with the rule that $i^2=-1$, and under the new system we can "add" splodges to weirdums to get new weirdums like $1+i$.

In solving differential equations we'd like a function which is zero everywhere except at a single point but which has a non-zero area under the curve. There is no function that behaves like this so we'll go to a larger system that contains not only functions but also spikes which do have the desired property because the larger system contains a rule about spikes which says they do. Conveniently certain calculations involving spikes cancel out leaving just functions.

A finitist or ultrafinitist shouldn't recognise the concept of infinite sets therefore the only sets are finite-sets and since all sets are finite the adjective finite is superfluous therefore from this point onwards we just use the term "set". Some people want to consider sets that contain things they haven't put in there themselves - which of course can't be done because a set only contains the items we've put there. So we embed the system of sets in a larger system that includes not only sets but also dafties. In this daft system the rules are that a daftie can have an "affinity" for things whether those things have been previously mentioned or not. Dafties have an affinity for things in the same way that sets contain things. A compatible embedding of a system of sets in the daft system means that a set has an affinity for the items it contains when the set is considered as part of the daft system, therefore by daft reasoning one can say things not only about dafties but also about sets. To each set one can attach a number. You can't do this with dafties so we embed the numbers in a larger system containing sinners and attach a sinner to each daftie. Sometimes there is a need for something that looks like a daftie but has no sinner - such things are called messes. A mess can have an affinity for collections of dafties that no daftie could have an affinity for. This could go on, but you need gobbledegook theory. The set system has zero gobbledegook. The daft system is level-1 gobbledegook. The messy system is level-2 gobbledegook. Finitists like to maintain a zero level of gobbledegook. Analysts are usually happy with one-level of gobbledegook and category-theorists are comfortable with any amount of gobbledegook.

There are two ways to compatibally extend a system:

1) a conservative extension adds new items but doesn't say anything new about the old items that couldn't be said before;

2) a progressive extension does say new things about the old items but only about things that were previously undecidable

P.S. We can combine the vapors, splodges and linedups in a system called the messysplodges but they haven't been studied much because they're a bit messy.