Actually, a more defensible version of formalism is worth looking at, deductivism. Here is a good summary:
The telling phrase for me is, “It is usually hoped that there exists some interpretation in which the rules of the game hold.” We might look at this as embracing the schism between “mere” validity and soundness.
Which war horses? Channelling the question into esoteric debates on the philosopy of math seems to be a bit of a red herring. Discovery and invention are plain English words with reasonably clear definitions. We don’t need to appeal to any authority to answer the simple question of whether math is invention or discovery, nor do we need to beat dead horses, we merely have to look at the definitions of the words and see where they apply.
“Discover” means “…to gain sight or knowledge of (something previously unseen or unknown)…” whereas “invent” means “…to originate as a product of one’s own ingenuity, experimentation, or contrivance…”
Are you seriously going to suggest that if there is one apple, and if another apple is set beside it, the result that there are now two apples is the product of somebody’s own ingenuity, experimentation, or contrivance?
Look at it this way: The steam engine wasn’t discovered. Absent its invention, it was not there to be uncovered by someone. Someone took principles of physics and utilized them to make a steam engine. Those principles of physics were not invented. The power of the vacuum and the power of expanding gas are extant whether humans realized they were there or not. To put it crudely, Aristotle would have enjoyed having his what’sit s***** whether he believed in the existence of a vacuum or not. If humans cease to be, and if enough time passes that all the steam engines in the universe rust to dust, it will still remain true that a vacuum has enough “power” (for lack of a better word) to lift mercury 30 inches at one atmospheric pressure. (Okay, bad science back there, but I only want to make the point.) The invention will be gone—no more steam engines—because no one is there to manipulate facts of the universe that exist whether someone is there utilize them or not.
If humanity ceases to exist, one plus one will still equal two. Bison will arrive at watering holes in line with the Poisson distribution. Rocks jettisoned from volcanos will still hit their maximum height only when their verticle velocity is zero. Rabbit populations will still match the Fibbonacci sequence. Pidgeons and rats will still have complete & transitive preferences. (I had to throw in one econ. tidbit.)
What won’t exist any more will be things like double-entry accounting, bid-call markets, black-jack strategies, and data encryption. Wave frequencies will still exist, but the algorithms for breaking them down to their component parts will cease to be. And just because there is no economist around to use Eigen vectors to predict input requirements for next year’s production, that doesn’t mean that Eigen vectors aren’t a natural part of our universe available for anyone with enough brains to once again uncover.
I suppose there is a lot of wiggle room here. Algorithms for doing long division are most certainly inventions. The notations of calculus are undeniably inventions. And so on. But these are all based on properties of the universe that existed before they came to be, and will exist after these algorithms are long gone. IIRC, Keith Devlin has a book called Mathematics: The Science of Patterns. Spheres can be packed only so tightly, and that is by a utilizing a particluar arrangement. Realizing that fact and uncovering the arrangement is an act of discovery. It is a scientific enterprise, so to speak. However, the method for doing long division is an invention, and it certainly falls under the mantle of mathematics, no? So I guess I have to change my answer and say that it is a mix. Much of math is the discovery of properties of the universe that exist regardless of whether we are there to observe them. That’s the discovery part. But there is a lot of mathematics, especially the stuff we learn in primary and secondary school, that is purely the creation of clever people for churning out answers to questions. That’s the invention part.
If one plus one will still equal two, then blackjack strategies will still exist. To the best of my knowledge, an optimal strategy (or set of strategies) for a game does not and never has depended on any instantiation of that game (but rather depends on the rules), so that if humans cease to be, blackjack strategies will persist, unless one and two also cease to be, in which case we are discussing your previous paragraph.
If our “one” and “two” are the same “one” and “two” that will persist after our death, then those entities and their relationships will also persist, in which case universal turing machines and fourier analysis will also persist, should those things be deductively linked to those numbers and their relationships blah blah blah. There will just be no one around to know them. If our “one” and “two” are not the same, then there is no question that mathematics will cease to be after we are gone.
No particular steam engine will exist, just as no particular game of blackjack will be played, but the design of any steam engine, by what I believe is the proper interpretation of your post, is not dependent on anyone knowing how to build one but instead depends on the laws of physics which, again, will persist after humans leave this mortal coil, if I am reading it correctly.
Who says there is? There are alternative set theories, but those that aren’t useful haven’t taken off.
IIRC, the axiom of foundation (which is a slightly more technical variation of the axiom that no set can contain itself as a member) is an axiom of ZF, and there are useful set theories that deny this.
Um, Platonism and Formalism. The rest of your post is mostly based on a misinterpretation, but there’s some that’s worth responding to. Basically, your assertions make sense, but not everything in philosophy is all it seems at the surface. Anyone can say that mathematical concepts exist independant of someone thinking of them, but how do you support that without resorting to “c’mon…”?
Very nice condensation, which makes clear how close that line of thought is to Platonism. If the “steam engine Form” exists independantly of any particular steam engine, then where is it? If it’s in something analogous to “Platonic Heaven”, how does this realm of Forms interact with the physical world? I could go on, but this line has been played out many times by better philosophers than I.
And, under the Structuralist view, Deductivism is a stunted form of Structuralism. It mimics the concepts of structures as far as possible without ever talking about structures themselves. Note the subtle point raised before:
To be more explicit: the “game” your cite speaks of is a structure, and the “rules of the game” are an axiomatic definition of that structure. The revolution is considering that structure as an object subject to study in and of itself. Please note: I don’t mean that it necessarily goes are far as reification, though Structuralist realists-in-ontology-and-truth-value would say it does.
I’m with erislover on this one, I think. Structuralism really just sounds like an object-oriented version of the same old debate.
Mathochist, is there a (somewhat) lucid yet comprehensive exposition of the various philosophies of mathematics? It can be a website or papers or book(s).
Wikipedia has a decent article, but the author is evidently mired in the 1930s. Really, he falls into the same trap of confounding Platonism with realisms and Formalism with antirealisms that I’m really complaining about. For the Structuralist view, though, I can find no more lucid explanation (and exploration) than Stewart Shapiro’s Philosophy of Mathematics: Structure and Ontology.
As I said, as regards the OP it only changes the venue from “Platonism/Formalism” to “realism/antirealism-in-truth-value”. As regards things like the Caesar and universality problems, it stands as separate from either. There are also plenty of other side effects I haven’t gone into; surely someone as steeped in logic as your handle would indicate can appreciate a viewpoint that puts ZFC, ZF!C, intuitionism, and so on all on the same (a priori) footing. Mathematics “based on ZFC” is itself a structure (more accurately, the topos of ZFC sets is a structure) as is mathematics “based on ZF!C” and so on. All the topoi stand beside each other, just waiting for the one that “really” obtains to be picked out.
Besides, unless you’re in your 70s you had your mathematical education post-Bourbaki, which was very Structuralist (though the label wasn’t used until later) and infiltrated everything done since. Frankly, you can trace New Math back to Bourbaki without much difficulty. What was first described to me as the Platonism/Formalism debate was a realist/antirealist debate with old names hung on the players. I have a feeling much the same is true for most people these days.
Actually, I wrote that from my own realist-in-ontology viewpoint. Antirealists-in-ontology (or at least eliminativists-in-ontology) would say that all the topoi are a priori equivalent, but only one “really” exists by virtue of being the description of the physical world.
Stanford, as ever, has an excellent entry on platonism which goes into some detail about different types, and tackles some nominalism along the way.
They have an entry on structuralism in physics, and I’ve read enough about structuralism in the social sciences (since this thread has started) to bore a librarian, but still nothing quite like structuralism in mathematics that offers an alternative to already-formulated metaphysics.
Finally, platonism is a type of realism, so obviously any discussion about platonism is going to be a discussion about realism.
I’m not sure how to parse this. Do you mean that nothing on Structuralism in mathematics you’ve read indicates an alternate “metaphysics”, or that you don’t see such in other fields? As an example of how realist Structuralist ontology differs from Platonism, for instance, Platonism says that the number 2 “exists” in and of itself, while realist Structuralism says that the number 2 only exists as a slot in a structure. There’s a 2-as-part-of-N, 2-as-part-of-R (R considered as a real algebra), 2-as-part-of-R (R considered as a real vector space), u.s.w.
Well, yes. Ultimately, I can’t answer the OP and I never claimed I could. I’m more trying to nitpick the identification of the two positions with Platonism and Formalism. The question is more about realism/antirealism of some sort (ultimately -in-truth-value) than about Platonism/Formalism.
Mathochist, it remains completely unclear to me what genuine alternative Structuralism offers as you’ve presented it. As with any abstract concept, there are various alternatives that have been neatly wrapped up in the so-called “Problem of Universals”, about which three general positions have been formulated: realism, nominalism, and conceptualism. Personally, I don’t think conceptualism is a genuine third option to what is otherwise a binary state (do universals exist or dont they?), but Ive never seen a very good defense of it, either, so I’ll admit that I might be missing something there.
For mathematics in particular, as seems to be common agreement, two positions have seemed to surface, which are platonism and formalism, which themselves have various sub-categories and nuances which aren’t necessarily worth elaborating on.
You have suggested several times that the platonic response is that “2 exists” and that’s that, and that structuralism offers an alternative by positing structures and not numbers per se, but I think it sells the position of platonism short to suggest that “2 exists” is the end of the platonist position. I almost feel like I’ve said, “Pyramids exist” and you respond, “All I see are squares and triangles, but I have this structure here that I’ll call a ‘shape holder’ where a pyramid-like thing fits…” :smack:
When we come to the rather definitive question of, “Do these mathematical structures exist”, you seem to suggest that we are stepping into a realist/anti-realist portion, and apparently structuralism finds a home for both positions. That being the case, it is not an alternative to either, but in fact a meta-theory, like relativism. Epistemological relativism, for example, makes no comment about whether objective knowledge exists, and so is just as happy in objectivism as it is in subjectivism. But that is simply because relativism is not an alternative to those ideas. Generally speaking, it seems that structuralism is a definitional/semantic framework which couches itself in some metaphysical terms but, upon inspection, takes no stance on them (i.e., it does not suggest that mathematical structures exist). But if it doesn’t answer that question, then it doesn’t resolve any issue about soundness of the reification of abstract objects like mathematical entities any more than some set theory resolved the issue of platonism by defining all natural numbers through zero and a successor function (is “2” real, or is the structure real, or is zero and a successor function real, etc, etc).
That is what I am not getting from you. How it is an alternative. There is nothing about platonism that demands “2” is some kind of object and is not the amalgam or consequence of other objects, though it is convenient to avoid more complicated maths and deal with arithmetic when discussing platonism. Nothing about platonism requires that “2” be a singular object, and nothing about platonism forbids that it be a composite one.
In case it wasn’t already abundantly clear, I’m not particularly well versed in the philosophy of math, but I find this thread very interesting and can’t resist the urge to continue posting in it.
I don’t think the bit about triangles and squares is really analogous. Mathocist isn’t saying that under structuralism the “pieces” of 2 exist but the whole does not. It seems that the structuralist point of view is that the relationships between 2 and other objects must exist, or at least that some object having those relationships must exist, but that it is pointless to try to define precisely what that object is. (I’m not sure “pointless” exactly captures what I mean, but I’ll go with it.)
The distinction Mathocist seems to be drawing is between something that “exists” and has “specific properties” and something that just has specified relationships but no definition beyond these. In structuralism, evidently all we can say is that there must be something called “two” that relates to these other objects in certain well defined ways, but it is impossible to know anything about two other than that it satisfies these relationships. Whereas, in platonism, at least as he has presented it, there is a real “two” that exists somewhere in “Platonic Heaven” and which has some sort of definition or list of properties beyond how it relates to the other natural numbers. This is why in platonism it makes sense to ask whether we’re talking about the Z-F two or the VN two, whereas in structuralism it doesn’t.
Whether you take the Platonist view to be something different is unclear to me. In particular, I’m not sure what you mean by a “composite” object. Again, I didn’t get the idea from what Mathocist was saying that “two” in structuralism was supposed to be built from other objects – rather, the “structure” in question was the set of relationships between the natural numbers, and “two” was just the name for whatever fits into the “two position” in this structure. If I’m understanding structuralism correctly, the structure of the natural numbers exists, but the numbers themselves do not, either as “singular objects” or “composite objects.”
I think Shapiro explains it better than I would paraphrase:
(underlines replace emphasis in the original, bold for emphasis mine).
Yes and no. I didn’t claim it was an alternative to realism or antirealism. There are realist and antirealist versions of Sturcturalism. I said it was an alternative to Platonism and Formalism. Realist-in-ontology-and-truth-value Structuralism is one version, antirealist-in-ontology-but-realist-in-truth-value Structuralism is another (closer in spirit to Formalism), and so on.
This gets into a hairy point in the literature. Classical Platonism does require that 2 be a Form in and of itself. Many writers, however, have used the term “platonism” as an unspoken shorthand for realism in ontology, whatever the ontology of the theory may be. One might turn your argument around and say that this sort of platonism is as much a metatheory (subsuming all theories which assert realism in their ontologies) as the general notion of Structuralism is. Alternatively, one might label the realist-in-ontology strains of Structuralism as “platonist Structuralisms”.
Ultimately, I think that it’s becoming clear (to me at least) that the main confusion is terminological (as it is so often). When I say “Platonism” I mean a very definite position. When you say “platonism”, you mean a class of positions which I call “realism in ontology” and which includes my “Platonism” as one of its “various sub-categories and nuances”.
Careful: To say that some object having those relationships must exist is to be a realist in ontology. To say that the relationships themselves must exist is to be a realist in truth value. Neither is representative of all Structuralists.
This is an excellent distillation (though I thought at this point it was better to err on the side of verbosity) of the content of my quote from Shapiro in my immediately prior post.
Or, for an antirealist in ontology, “given a collection of objects related to each other as following the Peano axioms, the conclusions of the theory of natural numbers follow. In particular, the object placed in the second slot will be the successor of the successor of the object in the zeroth slot, the predecessor of the object in the third slot, the first prime, and so on, where predecessor, successor, and prime are defined in terms of the relations filling the appropriate relational slots in the structure.”
I am willing to seriously suggest that the notion of putting one apple next to another apple and getting two apples is the product of the human mind. The human mind is marvelous at recognizing patterns, and deciding that two things are members of the same class. This task is very difficult to do on a computer, which may suggest that our brains have something built in to do it.
I think that a philosopher, or possibly a science fiction writer, could usefully consider the mathematics that an intelligent agent that did not naturally group things into classes might invent. If you put a Granny Smith next to a Pippin and said you had two “apples”, this hypothetical agent would think you were nuts.
I think that this is might be a third warhorse to add to Platonism and Formalism–mathematics that result from the structure of the human brain.
OK, so let me see if I’m understanding correctly:
A structuralist who is a realist in both ontology and truth value would say:
There is a certain “correct” set of relationships among the natural numbers, for instance the relationships defined by the Peano axioms, and there is some object that fills the “zero” slot, some object that fills the “one” slot, etc., but these objects only exist in so much as they fill those positions. It is meaningless to ask anything about “two” beyond its role in the structure.
A stucturalist who is an anti-realist in ontology but a realist in truth value would say:
There is a “correct” set of relationships among the natural numbers, and if objects exist that fill the positions in this structure, then we can say that they have the relationships given by this structure. For instance, if the correct set of relationships is the Peano axioms, then we can say that if objects filling the positions of this structure exist then they would have to satisfy the conclusions of the theory of natural numbers.
A structuralist who is realist in ontology but anti-realist in truth value would say:
There isn’t any “correct” set of relationships, but there really are numbers “zero”, “one”, “two” etc., and if they happen to fill the slots in a structure defined by the Peano axioms, then the conclusions of the theory of natural numbers will apply to these objects.
A structuralist who is anti-realist in both ontology and truth value would say:
We can’t say that anything exists, but we can say that if there is a “zero”, a “one”, a “two” and if they exist in a structure where their relationships satisfy the Peano axioms, then the conclusions of the theory of natural numbers will be true for these objects.
In any case, no structuralist would say that the objects exist beyond the role they play in their structure – so questions like “does two contain zero” are meaningless – whereas a platonist would say that the objects are real things on the same level as any other object, and questions that go beyond their role in the structure are perfectly legitimate.
Is this right? In particular, I’m a bit confused about the “realist in ontology but anti-realist in truth value” position. If in structuralism an object has no existence beyond the role it plays in the structure, then how can you have a real “two” but not a real structure with a “two”-position? Or are there no structuralists who are realist in ontology but anti-realist in truth value?