Am I a constructive mathematician?
- 02 October 2012
- Constructive math, General
It seems to me that people think I am a constructive mathematician, or worse a constructivist (a word which carries a certain amount of philosophical stigma). Let me be perfectly clear: it is not decidable whether I am a constructive mathematician.
But seriously, if anything, you may call me a mathematical relativist: there are many worlds of mathematics, and the view of the worlds is relative to which one I am in. Any attempt to bring mathematics within the scope of a single foundation necessarily limits mathematics in unacceptable ways. A mathematician who sticks to just one mathematical world (probably because of his education) is a bit like a geometer who only knows Euclidean geometry. This holds equally well for classical mathematicians, who are not willing to give up their precious law of excluded middle, and for Bishop-style mathematicians, who pursue the noble cause of not opposing anyone.
What could be more appealing to a mathematician than the idea that there is not one, but many, infinitely many worlds of mathematics? Would he not want to visit them all, understand how they are related, and see what happens to his favorite subject as he moves between them?
Let us consider an example. The real numbers are a mathematical object of fundamental importance, and have many aspects:
- The reals as a set are uncountable and in bijection with the powerset of natural numbers.
- The reals as an algebraic structure form a linearly ordered field.
- The reals as a space are locally compact, Hausdorff, and connected.
- The reals are a measurable space on which measure theory rests.
- The reals of non-standard analysis contain infinitesimals.
- The reals as understood by Leibniz contain nilpotent infinitesimals.
- The reals as Brouwerian continuum cannot be decomposed into two disjoint inhabited subsets.
- The reals are overt.
We can have some of these properties but not all at once. History has chosen for us a combination that is taught today as a dogma. Any attempt to deviate from it is met with opposition. Thus you probably consider 1, 2, 3, and 4 as true, 5 as something exotic you heard of, 6 as Leibniz's biggest mistake, 7 as intuitionistic hallucination (because obviously the reals can be decomposed into the non-negative and negative numbers), and 8 as something you never heard of (but you should have because it is the concept dual to compactness and you have been using it all your life).
Once we break free from Cantor's paradise that Hilbert threw us in we discover unsuspected possibilities:
- It may happen that the reals are in 1-1 correspondence with a subset of the natural numbers, while at the same time they form an uncountable set.
- It may happen that the reals form a proper class.
- It may happen that every real number has a Turing machine computing its digits.
- It may happen that the reals are not linearly ordered.
- It may happen that the reals are locally non-compact, in the sense that every interval contains a sequence without an accumulation point.
- It may happen that every subset of the reals is measurable.
- It may happen that the reals can be covered by a sequence of intervals whose cumulative length does not exceed $1$.
- It may happen that the reals contain nilpotent infinitesimals, which validate the 17th century calculations that physicists still use because, luckily, they did not subscribe entirely to the $\epsilon\delta$-dogma of analysis.
- It may happen that every real function is continuous, and consequently the reals are not decomposable into two disjoint inhabited subsets.
- It may happen that the reals are not overt, whatever that means.
It should be admitted that some of the possibilities are rather bizarre. For example, I do not know what good it is to have the reals as a subset of the naturals, but I am sure somebody could think of something. But why should a measure theorist ignore a world of mathematics in which every subset is measurable, or a computer scientist one in which all reals are computable, or a topologist one in which all functions are continuous, or an analyst one in which all functions are smooth?
I am not proposing that mathematics should be compartmentalized so that each branch sits in its world of mathematics, incompatible with others. That would be a grave mistake indeed. In fact, the unification of mathematics under the umbrella of classical set theory has been immensely successful precisely because it allowed mathematicians to discover deep and unsuspected connections between different branches of mathematics. We have learnt to look for connections between branches of mathematics, and now we must also learn to look for connections that span worlds of mathematics.
We cannot ignore the many worlds of mathematics. Therefore, mathematics must become applicable in a wide variety of worlds. Mathematicians have to be educated so that they develop multiple mathematical intuitions that help them feel how the worlds of mathematics behave.
I have so far not given you any technical definition of a mathematical world. Such a definition may be useful for showing meta-theorems, but I think it can never be exhaustive. A world of mathematics may be a forcing extension of set theory, or a topos, or a pretopos, or a model of type theory, or any other structure within which it is possible to interpret the basic language of mathematics.
At the moment I am visiting the Institute for Advanced Study as a member of the Univalent Foundations group. We are building a new foundation of mathematics whose language is type theory rather than set theory, and whose primary objects are homotopy types and not just bare sets. Do I think this is an exciting new development? Certainly! Will the Univalent foundations disrupt the monopoly of Set-theoretic foundations? I certainly hope so! Will it become the new monopoly? It must not!
(I think you mean "non-negative and negative numbers".)
You are not alone! However, the buzzword for people who hold these views is not relativism but pluralism. For example, <a href=http://www.mth.kcl.ac.uk/~davies/ rel="nofollow">Brian Davies</a> has written several papers about this, such as numbers 191 and 193 on his list.
Concerning your list of properties that the real line might have in different worlds, does anyone believe that it is not overt?
Thanks for reminding me about overtness, I included it in my list. Regarding "pluralist" versus "relativist", I think I want to stick with "relativist" even though I also like the other one. For me pluralism would mean acknowledgment of many worlds of mathematics, while I also want to make a claim about points of view. The mathematical method (or language, or a way of thinking) itself is part of a world of mathematics. For example, a type-theorist and a classical set-theorist use (somewhat) different formalisms and language. Consequently they each see the multiverse of mathematics from their own points of view, and this is where relativism comes in: there is no "absolute coordinate system" in the multiverse of mathematics. Views of the multiverse are relative to where we stand, and they are irreconcilable. Now that is a philosophical point of view!
@Ashley: yes of course, fixed, thanks. I shouldn't post things after midnight.
I wasn't arguing about the merits of the words “pluralist” versus “relativist”, just pointing out that if someone wants to <a href=http://www.google.com/search?q=pluralism+mathematics rel="nofollow">search the literature</a> for people with similar views then "pluralist" is the keyword that they need.
Andrej, I share your feeling that relativist/pluralist position is useful from a purely mathematical point of view, but would you acknowledge that certain views are more "in accordance with the real world", at least for particular purposes.
For example, some foundations may be more useful for thinking about computation, or physics or some other domain, while others may be completely un-useful for the same purpose.
Recently we were discussing something related with Mike Shulman, Ulrik Buchholtz and others on the nForum, when I kept asking (foundational layman that I am), questions like the following:
On the one hand I am very happy with the view that you are promoting: there is no good reason to exclusively consider some axioms and exclusively disregard others. All of them are mathematics.
On the other hand, I am wondering if there is not, after all, some joint common basis of all systems that we will recognize as being mathematics. Some absolute minimum of formal mathematical reasoning in which we are able to speak of any formal axiom in the first place!
Maybe the principles of "natural deduction" themselves? Before we specify any particular formation/introduction/elimination rules, but after we agree on what a valid set of formation rules should be?
For consider your example at the very end of the above post, concerning set theory and homotopy type theory: neither should have a monopoly, yes. But also: both of them you can implement in, say, Coq! So somehow there is something more foundational than both of them which subsumes them.
Ulrik and Mike eventually suggested that what I am after is the notion of "logical framework". I am not sure yet, mainly because it is not becomeing entirely clear to me what exactly people mean when they speak about logical frameworks, and if they even all mean the same thing. (Mike said that he may have carried part of this discussion to the IAS, so maybe you have heard an echo of what I am trying to ask here recently, I don't know).
While I feel strongly that I'd like to know the answer to what I am asking here, I feel just as strongly that I am probably not expert enough to formulate it properly. So if what I am saying here does not quite make sense to you, but if it reminds you of something that does make sense to you, then I'd be grateful if you could comment on that. Thanks!
It's not really about what's "in accordance with the real world". Formal systems are there to be plundered in any way you can. They don't come automatically equipped with a set of rules that says how to align a mathematical world with the real world so you can judge whether one is more in accordance with another. ZF is fairly poor at modelling bags of groceries despite the fact that sets are conceptually like containers, nonetheless I know how to use various theorems of ZF to predict how certain physical systems behave and hence convince my employers to pay me. Neither of these examples mean that ZF is more or less accordance in the world.
I think you meant "nilpotent infinitesimals", not "idempotent infinitesimals". See Todd Trimble's answer to: http://mathoverflow.net/questions/69569/various-flavours-of-infinitesimals
There is an extreme form of relativism in which mathematics is just a game played by upper class schoolboys and yah-boo-sucks to any relevance to the rest of the community.
In contrast to this is the (fairly conventional) view that mathematics is part of the Tower of Science, with Physics, Chemistry, Biology and so on built on top of it. I think this is the idea that Marc intended and which I feel Dan misrepresented.
I discussed that view in my <a href=http://www.paultaylor.eu/ASD/foufct rel="nofollow">Foundations for Computable Topology</a>, with the analogy that the role of Foundations is that of an Architect who is commissioned to provide a building for a Client such as Analysis, who in turn has Customers such as Physics. Physics has certain requirements of the Analysis that it uses (maybe compactness of $[0,1]$) but does not care how those are implemented in the underlying Foundations.
I was also thinking of the way in which the technology that has been used to provide components of our computers (screens and mass storage in particular) has completely changed during our careers, whilst remaining functionally equivalent.
On this analogy, I proposed in that essay that the Foundations should be designed for the mathematical discipline in question, starting from the Theorems that are characteristic of that discipline and used by its customers.
That proposal is pluralist in so far as it conceives of alternative ways of doing things, but still monolithic in so far as the disciplines together still have to support the Tower of Science.
Here is another analogy that might topple this monolith.
This view of mathematics at the base of the tower is a bit like the classical view of Biology in which the Cell with a nucleus (<a href=http://en.wikipedia.org/wiki/Eukaryote rel="nofollow">eukaryote</a>) is at the base of the tower of life (animals, plants and fungi). However, we now know that eukaryotes only form about half of the total biomass, the remainder being <a href=http://en.wikipedia.org/wiki/Prokaryote rel="nofollow">prokaryotes</a> (bacteria and archaea), which have increasingly been found to break all of the old rules of biology.
So maybe classical mathematics at the base of the tower of classical science is like eukaryotic life and the strange type theories that computer scientists invent are the bacteria and archaea of mathematics.
There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy., William Shakespeare, Hamlet (1.5.166-7).
@ Qwfwq: Fixed, thanks.
I understand your formalist perspective, and don't have a problem with it. I fully acknowledge that many forms of "useless" mathematics have found unexpected applicability after the fact. The freedom to explore the foundations of one's choosing, even if they aren't "useful", is a very good motivation for adopting the pluralist/relativist position being discussed.
But as Paul points out the formalist view competes in practice with what he is calling the "Tower of Science" that has THE mathematics as the bedrock of truth at its base. Paul's suggestion that we abolish the monolith and become more aware of different kinds of mathematics needed by different "customers" seems like a positive step to me.
As a happy user of many kinds of non-rigorous mathematics I'm not sure it can be said that I take a formalist view :-)
I like Paul's pluralist picture. I think it also suggests an interesting line of research. He suggests that Physics uses certain results like the compactness of [0,1]. Determining what axioms are needed by theorems is the subject of reverse mathematics. So I wonder if reverse mathematics should be extended to physics to find out what physicists need. (Unfortunately for this idea, real arguments in physics aren't always purely mathematical so it's not clear if this can be done usefully.)
Clearly the real numbers are a subset of the natural numbers since the natural numbers form a programming language, and the real numbers are simply those programs that output a stream of (signed) digits without stalling. :P
@Russell: because many programs compute the same real yor argument only shows that the (computable) reals form a subquotient of the natural numbers. To get an actual subset you need to pick canonical codes for reals, in a computable way, and that is not possible. So, if we are playing the computable game, you lose: there is no computable embedding of computable reals into the natural numbers.
Giovanni Sambin has developed something like this idea in his program of basic logic.
For my own part, linear logic and modal logics make me think this approach is likely to be very difficult. Since they do radical things to the context and the notion of hypothesis, I think any structural invariants capable of encompassing the variety of proof systems we know will be either very weak or very abstract. Noam Zeilberger (who will be at IAS for the homotopy year) has worked with Paul-Andre Mellies on this problem, so he can probably tell you many interesting things about this. (He has mentioned to me that there are connections between cut elimination and Isbell duality.)
Dan, thanks your your support for my idea of obtaining axioms from theorems, but please don't associate me with the phrase "Reverse Mathematics".
So far as I understand it, and I am willing to be corrected, this takes the framework of an $\in$ relation axiomatised in a first order way as given, and then switches the particular axioms on and off according as they are needed for significant statements in mathematics.
Following my computer component metaphor, this is like saying that the image on the screen has to be formed by a scanning electron gun, so that the job is to tweak the analogue circuits to do this better. In fact, that was my father's career, but he is dead now and we have a completely different technology.
In principle, therefore, everything in Foundations must be up for grabs. In practice, at a particular moment in history, we have to start from the (meta)technology that we currently have available. In <a href=http://www.paultaylor.eu/ASD/foufct/cat+type rel="nofollow">section 2</a> of Foundations for Computable Topology, I proposed, not introduction and elimination rules, or adjunctions, but the fact that these are equivalent, as the meta-technology.
You fixed 'idempotent' to 'nilpotent' in one place but not the other.
@Mike: I am amazed that you have read my blog post twice!
thanks, this is going in the kind of direction that I am after, yes.
To make this more concrete, maybe we can create a table displaying part of the "hierarchy of foundations", starting with the most basic natural deduction framework/basic logic (or whatever), and successively adding axioms.
I gave it a try on this page. This is to provoke reactions (on your parts) and generate insight (on my part) not to claim anything.
I somehow feel that the major problem here is one of equivocation. Naturally, most mathematicians will balk at you announcing "The interval [0,1] is not compact," (even if you precede that with "I have found a framework in which..."). On the other hand, saying "The pseudointerval [0,1] is not compact," will result in a much more amiable response, even if no one has any idea what a pseudointerval is and how it relates to the standard interval. The mathematical content is the same, but the social (or linguistic, if you like) content is vastly different. I also don't think this only happens in interactions between multiple mathematicians. In fact, I'm sure that if there were a single Mathematician and he were considering intervals and pseudointervals simultaneously, he would become very confused very quickly if he did not start making some distinction between them (in his mind or wherever).
@Miha: Ah, but I am an example of someone who can hold two intervals in his head, and not get confused. Of course, the interval cannot be compact and non-compact at the same time. But there can be two worlds in which there are two intervals, one of them being compact and the other not. And both objects have an equal right to be call "the interval". But you are yourself quite familiar with a situation which is not much different. Just consider the fact that $\aleph_1$ may be this or that, depending on what model of set theory you're in. Both deserve to be called $\aleph_1$, but they may have different properties. I don't think it make sense to brand one of them as being somehow "pseudo".
Miha, I don't think that just the "proper" interval should be called the interval and others prepended with some modifier. At the risk of over generalizing, the situation doesn't seem different than all the other identifications that mathematicians make all the time, for example with different notions of continuity in different theories. The disambiguation then comes from the context in which the expression occurs and this should be no different when considering more foundational issues. A mathematician shouldn't be shocked when presented with things called intervals which don't have property P dear to him, just as he isn't when compact sets turn out not to be closed.
Andrej, no, neither of them deserves to be called $\aleph_1$, or rather neither of them deserves to be called just $\aleph_1$, precisely because they may have different properties. People place all sorts of decorations on their cardinals to avoid this sort of confusion. Since the relativist position cannot choose a preferred canonical model (or framework), these distinctions are necessary. The same would go for the interval example.
Ales, I agree that mathematicians have no trouble identifying isomorphic objects. The trouble is that we are now asking them to identify objects with very different properties. They may be the same in some respects, and the people who only care about those properties will happily go along with you, but the other ones will struggle. Of course, if they are familiar with the context this will be easier. But these issues arise even when introducing the context and there another way has to be found. And I disagree that mathematicians shouldn't be shocked when a term they know is somehow subverted. I see it as entirely reasonable for a person who's dealt with metric spaces all his life (but hasn't seen a topological space, say) to be outraged if I come along and start calling some non-Hausdorff thing a metric space (even if I've just slightly weakened his definition of a metric).
I've been trying to figure out whether the names mathematicians use for their concepts are rigid or definitional or what. The fact is that concepts carry all sorts of baggage (the interval is not only the set of points between 0 and 1, but is also compact, connected, a dense order etc.), even if people don't admit it, and this has to be taken into account.
Miha, are you outraged that there could be triangles whose angles don't add up to $180^\circ$ or that $-1$ could have a square root?
Are you going to add codicils to the obvious definition of a triangle to prevent Riemann, Bolyai, Lobachevsky and Einstein from subverting your preconceptions about geometry?
Why should computer scientists and algebraic geometers whose spaces are not Hausdorff accept your codicil requiring compact subspaces to be closed? (Or, indeed, Bishop's that they should be overt?)
As for metric spaces, the symmetric axiom is not needed for the essential development and would seem to be a matter of prejudice. <a href="http://www.lix.polytechnique.fr/Labo/Eric.Goubault/" / rel="nofollow">Eric Goubault</a> has shown how to develop directed homotopy theory and apply it to computational processes.
The relativist or pluralist ideas that are controversial now will be accepted as part of the mainstream in due course, exactly as non-Euclidean geometry was by the end of the 19th century.
There is an economic reason for this: Non-classical mathematics is useful in computer science, which is useful to the economy and so attracts funding, which classical pure mathematics increasingly does not. Eventually this message will get through to the best 18 year old mathematicians, who will choose to study modern computer science instead of fuddy-duddy mathematics. Then the latter will die.
Nor is this new. The tradition of mathematics has gone under many names (geometry, astronomy, ballistics) in the past, so it will come to be called computer science in the not too distant future. "Mathematician" will once again mean an astrologer.
@Miha: Now that you got a deserved beating from Paul, let me just point out that it makes very little sense to identify two objects which live in different universes. So this has nothing to do with confusing two different concepts. There is $\aleph_1$ in this model and $\aleph_1$ in that model. Both are defined as the least uncountable ordinal. This is a valid definition because it fixes the set $\aleph_1$ up to a unique set (I am assuming standard ZF-style treatment of ordinals). What we have to get over with is that a definite description might fix an object but it does not fix its relation to other objects. This should be obvious, as the "other objects" depend on what universe we are in. The whole point of forcing is to produce other models of set theory. With the millions and millions of set theory models running around, all of them equally plausible, how could anyone pretend that one of them is special?
Sorry, we should be more welcoming to visitors to our playground and not bully them like <a href=http://www.cs.nyu.edu/pipermail/fom/2009-June/013781.html rel="nofollow">FOM does to us</a>. (Not sure if this the the right reference.)
A snappier definition of $\aleph_1$ is the set of isomorphism classes of well-orderings of subsets of $\Bbb N$. This is meaningful in Higher (maybe even Second) Order Logic but is obviously contingent on what predicates (subsets, relations) there are in the system. How anybody could expect this to be universal, or even relevant, escapes me. I really don't think it belongs in the Tower of Science.
PS I had even forgotten what $\aleph$ looks like!
Political correctness has no place in mathematics. Would Gödel have been more productive as a "pluralist" than a Platonist? No; as he explained, his Platonism was the reason he was able to discover incompleteness. If there exists mathematical truth, then pluralism is not augmentation - it is dilution.
@Ivan: Part of my relativist position is, and this I want to stress, that the relative points of view cannot be reconciled. One possible world is Platonism. From that world my relativism looks like model theory, with the additional (false) claim that models are just as good as the ideal world of mathematics. Notice that we are not talking mathematics or logic here, because we are saying that some world/models are "more real" than others. I am not advocating that all mathematicians should adopt my view, as that of course would be a form of absolutism. I still can believe that my view will be prevalent in due time (for economic reasons pointed out by Paul), and I took the liberty of admonishing Miha because I have known him personally since he was just a little mathematician boy.
@Ivan: Platonism is a philosophical position with respect to ontology. Gödel proved an epistemological fact: some truths aren't knowable. What is the difference between a truth that exists and one that doesn't if you can't possibly know it?
Different foundations posit both different ontologies (what kinds of mathematical objects exist) and different epistemologies ( proof systems ). How are you going to distinguish the relative "mathematical truth" of one or the other?
I personally privilege computational truth (which is somewhat objective) as the basis for mathematics, but I have to acknowledge that this is a philosophical (and practical) decision that lies outside of mathematics per se.
Ivan, I don't know what you mean by Political Correctness in this context, but surely before Pluralism takes on the technical meanings that we have been discussing, it first means that the world is a richer place through having people with many different points of view in it.
I fully understand that a mathematician who sees himself "at the coalface" or following his ancestral instincts to hunt wild animals will find it easier to do so by believing that the world is real and chasing down his counterexample to kill it. Indeed, Lobachevsky and Bishop would not have achieved what they did without a pretty thorough grounding in classical geometry or analysis.
However, the fascinating and unexplained thing about mathematics is that the arguments survive even when you turn the world upside down by swapping theorems with definitions or taking the basic intuitions away. For example, a polynomial has a multiple zero iff both it and its derivative vanish -- even in a field of finite characteristic, making limits and rates of change meaningless.
The arrogance of classical mathematicians is not just that they think that their world is the only real one (despite plenty of evidence to the contrary since Godel and Cohen) but that their accounts of the arguments are the definitive ones.
Intuitistic mathematicians do not "rob" their classical colleagues of their theorems, as Hilbert alleged, but nuture and polish them. Often there are many intuitionistic notions (for example with <a href=http://www.paultaylor.eu/ordinals#intso rel="nofollow">ordinals</a>) or theorems where there was only one classical one. Andrej is better able than me to give you examples of things that only exist in delicately constructed intuitionistic worlds.
"Part of my relativist position is, and this I want to stress, that the relative points of view cannot be reconciled. One possible world is Platonism."
How do you mean "possible world"? The pre-eminent theoretician of possiblilia was David Lewis, and, according to him, all possible worlds have the same abstract entities. It seems incoherent to call Platonism a "possible mathematical world" - if Platonism is true, then there are no other possible mathematical worlds.
"Platonism is a philosophical position with respect to ontology. Gödel proved an epistemological fact: some truths aren’t knowable. What is the difference between a truth that exists and one that doesn’t if you can’t possibly know it?"
First, that the "truth" that doesn't exist doesn't exist, and the one that does exist does exist. That is a clear difference. Second, a "truth" that doesn't exist is not true.
"Different foundations posit both different ontologies (what kinds of mathematical objects exist) and different epistemologies ( proof systems ). How are you going to distinguish the relative “mathematical truth” of one or the other?"
Such considerations would have to be based on other philosophical positions. However, if you privilege one, as you privilege computation, then you cannot coherently believe in relativism.
"The arrogance of classical mathematicians is not just that they think that their world is the only real one (despite plenty of evidence to the contrary since Godel and Cohen) but that their accounts of the arguments are the definitive ones."
How can more than one world be real?
Andrej, I agree that it makes no sense, in most cases, to identify objects from different universes. However this is precisely what happens if I'm not careful with my terminology. If I stop making these (I suppose linguistic) distinctions, then there simply is no context from which you can infer what I'm actually talking about.
Perhaps you can comment on the following scenario: starting with a model of set theory, I force to collapse $\aleph_1$. What do I call things in the extension? In particular, at what point and how does my old $\aleph_1$ lose the right to be called that? Slightly more tongue-in-cheek, I ask you, with the millions and millions of models of PA running around, all of them equally plausible, how could anyone pretend that one of them is special? And yet we talk about such things as true arithmetic. (I hope you don't mind too much that I seem to have somehow provoked some kind of argument on your otherwise very presentable blog. Also, are you implying that I'm no longer a little mathematician boy?)
Paul, I am outraged that there could be a triangle whose angles don't sum up to 180°. However, this is because to me (and most of the mathematical community, I should think) a triangle is a certain arrangement of straight line segments in the Euclidean plane, where the angle thing is just a theorem. To be clear, I have no problem with the fact that there are object which are very similar to triangles in certain respects, but the sum of whose internal angles isn't 180°. What I do have a problem with is freely calling these things triangles. I think statements of the type "There is a triangle whose angles don't sum up to 180°," are wordplay and what mathematical content they posses could have been conveyed much more clearly and, dare I say, correctly.
On the other hand, I admit I don't really care what people call their concepts among themselves. I will, however, insist on some sort of accommodation if they want to talk to me about them.
Also, the doomsaying about the death of classical mathematics. A bit strong, don't you think?
@miha: making another comparison to your suggestion that we should call intervals in other models “pseudointervals”, how would you feel about the statement “2 is prime”? It is true in Z, but not in e.g. the Gaussian integers. I am confident that most mathematicians, asked “Is 2 prime?” would at first say “yes”, since Z is the default setting for these concepts, but that when reminded of the Gaussian integers, they would happily agree that in other rings, 2 may or may not be prime. They would not feel the need to say “pseudo-2” or “pseudo-prime”; they would just say that the statement “2 is prime” can be interpreted in multiple ways. That does not make “2 is prime” an imprecise statement: as long as one stays within the language of rings, one can reason with it completely rigorously, without fixing a particular interpretation. Nor does it mean they are rejecting platonism — even if there is one true mathematical world, there are certainly many rings in it.
For the full language of set theory, most mathematicians are much more wedded to a single default interpretation than in the case of rings. So I think I agree with your point that sociologically, it might sometimes be a good idea for us to say “pseudointerval”, just to emphasise that we are considering a wider range of models than they may be used to. However, if we choose instead to use the term “interval” analogously to how most mathematicians use “2” or “prime”, it is not wordplay; it is certainly not imprecise or incorrect; it is not even rejecting Platonism.
Personally, I think (though I’m not sure) that I’m something both of a Platonist and a relativist: I feel there may be one true world of mathematics, but I certainly do not think there is just one true way of reasoning about it! A model of set theory (or any other foundation) is just a certain way of using the language of sets to describe a certain part of the true world of mathematics, looked at from a certain point of view. There may be one true world of mathematics; but it does not contain a distinguished “true” model of set theory any more than it contains a distinguished “true” ring.
Paul Taylor wrote:
Non-classical mathematics is useful in computer science, which is useful to the economy and so attracts funding, which classical pure mathematics increasingly does not.
Can you point me to some document that substantiatiates this statement with empirical studies, with numbers or the like?
(I don't doubt the truth of the statement! But all the more would it be nice to have a proof in case I run into somebody who does doubt it. If you have some pointers, I'd be grateful.)
@Ivan: I think I mispoke there a bit (but only a bit) when I said that "Platonism is one possible world". You see, to me Platonism is one possible world, but of course to a Platonist that is non-sense. Which is why my philosophical position is a position, not just an idle remark that everyone can agree with. The question then is how to "speak across the worlds" (something that I find extremely interesting). To speak to a Platonist, I would have to water it down by saying "Platonism is one of the views of mathematics that people hold", which establishes the fact that there are many worlds of mathematics only in the sense that people have opinions (which can therefore be false and mistaken, except of course for the Platonist one). So if you are happier with my saying "personal view of mathematics" instead of "mathematical world", read it that way. To put in derogatory terms: to a person who can see the other worlds Platonists are like two-dimensional beings that deny, or cannot comprehend, the existence of the third dimension; consequently it is difficult to discuss certain topics with them. By the way, I think your pronouncements on existence and non-existence of truth make no sense, especially when you equate "non-existence of truth" with "falsehood".
@Miha: I think you are discovering the imprtance of speaking in context. When I say "the cow" you need some context to know which one I mean. It is exactly the same with $\aleph_1$. And that is all there is about it, becaues the same argument applies to triangles. There is such a thing as a universe of discourse, which matters, especially when we switch from one universe to another. Of course when you speak about several models at once you will need to distinguish between this and that $\aleph_1$, but they are both still just $\aleph_1$, each in its own world. All this is a bit complicated and difficult to disentangle because humans are used to the context being implicit. We usually guess the context in which communication should happen, and to a large extent connected communities of people share a common context that is good for most purposes.
You also asked about "the standard model of PA". I think you can view this in two ways. The easy way out is to say that "the standard model of PA" is just some unspecified model in which we are working. But this really just circumvents the question. I think it is evident that we cannot really know which model of PA is the standard model, nor can we know whether "your" model is the same as "mine". In fact I don't think either of us actually holds a model of PA in his mind, at least not in any real sense. Perhaps as some sort of a higher-primate approximation of the thing that a model is supposed to be (although you have the advantage of being a couple of inches taller than I am, so not a boy anymore). My solution is to admit that there are in fact millions and millions of models of PA, all slightly different, but informal mathematics does not happen in any one of them. Informal mathematics is a human activity, of which formal logic is an approximation (or the other way around, if you are a die-hard logician with poor social skills).
Perhaps I should think more carefully about a relativist position with respect to mathematics as a human activity vs. relativist position as the philosophical position of admitting that there are many equally valid "realities" of mathematics (if you just saw scare quotes you are a Platonist). On the other hand, I can probably explain everything using relativism. Some people will view mathematics as a human activity, and they would think that is what I am talking about, while others will consider this discussion to be about actual mathematics, and will undertand "mathematical world" as topos, or model of ZF, or a logic, etc. I would say that the pluralism of human mathematics is being reflected by its mathematical idealization.
@Urs: If I am reading correctly the data in this NSF report, see for example Table 19, computer science and mathematics together get a miniscule amount of money compared to life sciences, the ratio is 1 to 10. Then within the CS+math clump CS gets 2.6 as much money as mathematics. Now suppose 10% of computer scientists "do math". This would then account for roughly a quarter of all the money that mathematicians get, so "CS math" is an important "branch" of mathematics. And I am not even considering private funding, of which CS gets a lot and math very little. But I think "CS math" is still smaller than "applied math", although I don't have data on that. If following the money is the right thing to do, smart young mathematicians should be doing biology and medicine.
Speaking as a computer scientist, I certainly hope this statement is false! IMO, the contribution of computer science to logic is that it has helped us finally take a step out of Frege's shadow. That is, I see Frege's invention of predicate logic as a bit like the development of photosynthesis: it was a stupendous advance, which triggered a mass extinction of a variety of other conceptions of logic. Things that had been part of logic, like causality, intention, and inductive inference, went into eclipse since our old methods could not compete with the sophistication of the model-theoretic, truth-functional approach. Intuitionism, plus Gentzen and Prawitz's proof-theoretic approach, plus their applications to computer science, at long last give us alternative conceptions of logic strong enough and useful enough to stand as an alternative to the traditional approach.
But I dearly hope that this is not the end of the story! Seeing how incredibly profitable the notion of construction has been, I wonder how much more we can enrich our notion of logic if we allow causation and dialectic and so on back in. (Heck, even Frege still offers us obviously-important ideas that we lack a good formal model of, like the distinction between sense and reference.)
I find your perception that this lack exists surprising. My own informal sense of this is that you have "sense" when you have a constructive process to recognize a unicorn if you found one, and "reference" is being able to find a unicorn to be recognized by the sense. What is challenging to account for, given that formulation?
@Andrej: I just wanted to applaud your response to Martin Davis in that FOM exchange from three years ago that was linked to by Paul (which I've only just seen; I don't tune into FOM with any regularity). It was spot-on in every respect.
While I am sympathetic with the idea that there are many meaningful ways to consider e.g. the continuum, this view has, as far as I see, the same problem as any relativistic/pluralistic account: Even if there are many "mathematical worlds", there must still be something that they have in common and that entitles us to call them "mathematical". This something then provides a single, unifying foundation of mathematics that we should strive to grasp and describe. The pluralism can in this context take a methodological function by preventing us from assuming that our current working notion is ultimate and cannot be corrected or improved any more.
@Merlin: you are almost hitting the gist of my position. I am precisely attacking the assumption that "there must still be something that they [mathematical worlds] have something in common that entitles us to call them mathematical". No, that restores absolutism. Instead, I put forward the thesis that each mathematical world has its own idea of what all the worlds have in common. Of course, this leads to the (quite realistic) possibility that the various mathematical worlds disagree about what the various mathematical worlds are. What counts as mathematics in one world might not in another! For example, some mathematicians would refuse to call "mathematics" a development of mathematics based on paraconsistent logic (and some do). Or some mathematicians would declare unfettered use of impredicativity meaningless, and so they would declare many kinds of mathematics as meaningless (I know some who are at least suspicious of impredicativity). I see no reason why such a position is problematic. As far as I can see it does not follow from this that "anything goes". Actually, it problably does, but only in some worlds. I am hopelessly relativist, and I do not have a problem with it. I do have a problem with the assumption that mathematics is absolute. It is not. It is a man-made artefact ridden with historic accidents. When the aliens land we will be quite surprised that they have mathematics which is nothing like what humans ever imagined it to be.
However, giving up on a relativist position as unanalyzable or unmathematical is like giving up on studying infinity on the presumption that it is somehow unanalyzable. A challenge is presented here. How do we make sense of the fact that there are many mathematical worlds, without an absolute common basis? If phsycists could do it, we can do it too.
just wanted to applaud your response to Martin Davis in that FOM exchange from three years ago that was linked to by Paul (which I’ve only just seen; I don’t tune into FOM with any regularity). It was spot-on in every respect.
[…] “classical” mathematics: they simply are two different worlds, as explained very nicely by Andrej Bauer. The universe of mathematics would be poorer without any of them. Similarly, the universe of […]