What does 2 + 2 = 4 mean?

[Architeuthus]

My views are probably best classified as a form of Aristotelian realism, so they fall into a somewhat odd middle ground between Platonism and empiricism.

I'd say that 2+2=4 follows upon the observation that whenever a grouping of any two objects is conjoined with a grouping of any other two objects, then a grouping of four objects altogether results. The numbers 2 and 4 would be concepts of quantity, or “manyness,” where 2 conceptually denotes 2-manyness, and 4 conceptually denotes 4-manyness, and concepts of quantity would be derived from instances of it observed to occur in the world.

Thank you very much for replying, and for that link.

"Aristotelian, or non-Platonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are."

That does sound exactly like what I was calling empiricism, although "The objects may be of any kind, physical, mental or abstract" and "An essential theme of the Aristotelian viewpoint is that the truths of mathematics, being about universals and their relations, should be both necessary and about reality" makes me hesitate.

Let's pick a ridiculously large number like the prime P = 2^43112609 − 1. What do you think P + P = 2^43112610 − 2 means, exactly?

And what about infinite sets? Franklin seems to suggest that that's where the difference with Platonism kicks in, but he doesn't explain himself (although he hints at tossing out some of the traditional mathematics of infinite sets).

 

[Messy]

Other, it just means that two pieces of cake plus two other precies of cake makes a whole bunch to eat.

 

[Architeuthus]

Other, it just means that two pieces of cake plus two other precies of cake makes a whole bunch to eat.

That's just my "Empiricism" option made a little vague.

AND I CAN SEE WHY YOU MADE IT VAGUE! THERE ARE ONLY THREE PIECES LEFT! YOU ATE ONE!

 

[Crandaddy]

Thank you very much for replying, and for that link.

You're very welcome. It's my pleasure.

"Aristotelian, or non-Platonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are."

That does sound exactly like what I was calling empiricism, although "The objects may be of any kind, physical, mental or abstract" and "An essential theme of the Aristotelian viewpoint is that the truths of mathematics, being about universals and their relations, should be both necessary and about reality" makes me hesitate.

It's debatable to what precise extent Aristotle's views differed from those of his teacher. He really didn't so much dispense with Plato's Ideas as pluck them from their heavenly abode and bring them down to earth, so to speak.

Aristotelianism is distinguished from Platonism principally by its emphasis on empiricism, but even so, it's not exclusive of uninstantiated formal objects (such as mathematical objects or universals), or of the possibility that we can know necessary truths about them. Aristotelianism typically holds that such objects can't exist as entities all by themselves, but it doesn't insist that they can't be uninstantiated in material substances.

Let's pick a ridiculously large number like the prime P = 2^43112609 − 1. What do you think P + P = 2^43112610 − 2 means, exactly?

It means that from our observation of patterns, relations, etc. in the world we can discern that there is a formal reality expressible as “((2^n) − 1) + ((2^n) − 1) = 2^(n + 1) − 2” that determines how the world necessarily has to be structured. We learn about that formal reality by way of empirical observation, but I (as well as other Aristotelians) hold that we learn necessary truths about how the world has to be structured by abstracting from our observations of how it actually is structured.

And what about infinite sets? Franklin seems to suggest that that's where the difference with Platonism kicks in, but he doesn't explain himself (although he hints at tossing out some of the traditional mathematics of infinite sets).

Platonists tend to view mathematical objects as entities in their own right, so a typical Platonist would have no qualms about saying that infinite sets (such as numerical sets) actually have infinitely-many members.

Aristotelians, on the other hand, generally don't have any tendency to either affirm or deny that there actually are any infinitely-membered sets. Some might lean one way or the other, but not because of any views that are distinctly Aristotelian. But if that's the case, then how could an Aristotelian account for, say, the seemingly infinite set of positive integers without committing himself to the actual existence of infinitely-many of them? He could do it by saying that the members of a set can be infinite, not in actuality, but in potentiality -- not by virtue of the unlimitedness of the set's actual membership, but by virtue of its unlimited potential to increase in membership. And so he could say that sets we consider to be infinite (such as the set of positive integers) are thus potentially infinite if they're not actually so.

 

[Architeuthus]

It's debatable to what precise extent Aristotle's views differed from those of his teacher.

Yes, but what Franklin is calling "Aristotelian realism" doesn't seem to be quite the same as what Aristotle said (indeed, Franklin seems to be the main advocate of mathematical Aristotelian realism).

It means that from our observation of patterns, relations, etc. in the world we can discern that there is a formal reality expressible as “((2^n) − 1) + ((2^n) − 1) = 2^(n + 1) − 2” that determines how the world necessarily has to be structured.

I think you misunderstood me. If P = 2^43112609 − 1, the universe contains far fewer than P particles, so the number P can't be instantiated in material substances (Franklin calls it an "uninstantiated universal"). Neither, according what you say, can it exist as an entity all by itself (that would be Platonist). So what (if anything) does it mean? How do we interpret statements about its being prime, or statements about the value of P + P? The same problem is true in spades for infinite sets.

Aristotelians, on the other hand, generally don't have any tendency to either affirm or deny that there actually are any infinitely-membered sets. Some might lean one way or the other, but not because of any views that are distinctly Aristotelian. But if that's the case, then how could an Aristotelian account for, say, the seemingly infinite set of positive integers without committing himself to the actual existence of infinitely-many of them? He could do it by saying that the members of a set can be infinite, not in actuality, but in potentiality -- not by virtue of the unlimitedness of the set's actual membership, but by virtue of its unlimited potential to increase in membership. And so he could say that sets we consider to be infinite (such as the set of positive integers) are thus potentially infinite if they're not actually so.

This is why I was frustrated by the vagueness in Franklin's paper. It seems to me that the Aristotelian point of view requires either tossing out infinite sets, or following the constructivist approach of tossing out uncountably infinite sets. The countably infinite sets (like the integers) can be given an interpretation as infinite in potentiality, because they are the limit of a sequence S1, S2, S3, ... However, Franklin doesn't seem to like the price you have to pay for being constructivist.

Franklin's book (which I have glanced at online but not read) seems to articulate and go for what he calls "semi-Platonist Aristotelianism" (which does allow infinite sets). He views that as very different from classical Platonism, although I think he's misrepresenting classical mathematical Platonism, so that the difference is much smaller than he thinks.

Also, his discussion of whether uninstantiated universals "exist" seems to me to dodge the question ("it is not the business of the philosophy of universals or the philosophy of mathematics to deal with complex questions in the philosophy of language concerning reference to objects beyond the here and now" -- that's simply a refusal to answer the $64000000 question I'm asking in this thread).

 

[Crandaddy]

I think you misunderstood me. If P = 2^43112609 − 1, the universe contains far fewer than P particles, so the number P can't be instantiated in material substances (Franklin calls it an "uninstantiated universal"). Neither, according what you say, can it exist as an entity all by itself (that would be Platonist). So what (if anything) does it mean? How do we interpret statements about its being prime, or statements about the value of P + P? The same problem is true in spades for infinite sets.

Maybe there's a real formal realm that exists necessarily (uninstantiated in some cases), and our minds are able to access it when we make such statements. This does, of course, run us very close to Platonism (and perhaps even a bit closer than Aristotle would have liked), but even so, we still don't have to go full-Platonist: We can still say that those forms can be instantiated in material substances that we can perceive with our physical senses. We can also still say that if they're not instantiated in material substances, then they exist as ideas in minds, and even if they don't exist as ideas in any contingent minds, then they exist necessarily as ideas in the mind of God (which would free us of having to say that they exist as entities by themselves, independent of mind and matter).

This is why I was frustrated by the vagueness in Franklin's paper. It seems to me that the Aristotelian point of view requires either tossing out infinite sets, or following the constructivist approach of tossing out uncountably infinite sets. The countably infinite sets (like the integers) can be given an interpretation as infinite in potentiality, because they are the limit of a sequence S1, S2, S3, ... However, Franklin doesn't seem to like the price you have to pay for being constructivist.

Well, why can't we discover from our empirical observations that it is the nature of objects in the world to be quantifiably expressible according to infinite sets whose membership exceeds one-to-one correspondence with the set of natural numbers, so that for any member of such a set that might ever be discovered, there is some subset of possible worlds in which it is expressed.

Of course, it would be the case that members of such infinite sets can only be discovered (by us) sequentially and in one-to-one correspondence with the set of natural numbers, but it seems to me that all that means is that such is the nature of our thought that we're only able to discover potential instances of transcountable infinities along countably sequential lines. I don't see why it should mean that we can't discover, from our observations of the actual world, the real potential for there to be infinities that exceed countability, in much the same way as the physicist (for example) discovers, from his observations of the actual world, the real potential for natural bodies to move and change -- those potentialities (in both cases) having a formal existence in the mind of God, if nowhere else.

 

[Architeuthus]

Maybe there's a real formal realm that exists necessarily (uninstantiated in some cases), and our minds are able to access it when we make such statements. This does, of course, run us very close to Platonism

You're not kidding.

But mathematical Platonism doesn't take on board all of Plato, it merely commits to Hardy's dictum "that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations,' are simply our notes of our observations" (where the "mathematical reality" is taken to be some non-physical domain).

We can also still say that if they're not instantiated in material substances, then they exist as ideas in minds, and even if they don't exist as ideas in any contingent minds, then they exist necessarily as ideas in the mind of God

That takes you to mathematical Platonism as many Christian philosophers would express it. To quote Kepler "Geometry existed before the Creation, is co-eternal with the mind of God, is God himself (what exists in God that is not God himself?)" (i.e. taking ideas in the mind of God as being part of God).

I believe that Aquinas (Summa I:16:7) also said "The nature of a circle, and the fact that two and three make five, have eternity in the mind of God."

Well, why can't we discover from our empirical observations that it is the nature of objects in the world to be quantifiably expressible according to infinite sets whose membership exceeds one-to-one correspondence with the set of natural numbers, so that for any member of such a set that might ever be discovered, there is some subset of possible worlds in which it is expressed.

Yes, sure. One can imagine possible universes where the number of particles exceeds any given natural number N. But that leads to the question: in what sense do those possible universes exist?

And there are tricky issues associated with uncountably infinite sets relating, not to their discovery, but to the idea of an infinite set being the limit of a finitary process. I can't see how an uncountable set can be the limit of a finitary process (but, naturally, I might be missing something).

Of course, mathematical Platonism solves all those problems.

those potentialities (in both cases) having a formal existence in the mind of God, if nowhere else.

Once you say "formal existence in the mind of God," I think you've pretty much committed to mathematical Platonism in its Christian form. You and Kepler could have sat down over a beer in Linz and have been 100% in agreement, it seems to me.

 

[nonbeliever314]

Platonism to some degree with me. I'm with Max Tegmark on this one. MUH.

 

[Architeuthus]

Platonism to some degree with me. I'm with Max Tegmark on this one. MUH.

I really should have listed MUH as a separate option, because it's much more radically Platonist than the usual "mathematical Platonism." Tegmark, if I understand him correctly (and I must admit, I haven't read the book) doesn't just separate the mathematical and the empirical, but actually makes the latter subservient to the former.

 

[nonbeliever314]

I do think the universe is purely mathematical. How? Not sure, but I think it has something to do infinity (long story). Why is it mathematical? Doesn't matter. Evidence to support the claim? Nothing besides the patterns we see in nature. Conclusive evidence? Wish me and whoever else luck. If it turns out not to be the case, i'm happy. If it turns out to be the case, still happy.