1 + 1 = 2 is true by definition. I’m sure you know that in number theory, the successor of 0 is 1, and the successor of 1 is S(1), which is what we call 2, and the definition of addition depends utterly on these. Bertrand Russell was a bit dismayed to find that pretty much, all of math ends up being tautological. To some extent, that’s true, but it doesn’t make it any less useful.
The fact that “1 + 1 = 2” is true by definition doesn’t mean it applies to anything, in the sense that “1x + 1x = 2x” for any x, because it’s not true if “x” is a velocity. However, if you do the detailed analysis, those end up being two different x’s, because one is “velocity of A wrt C” and the other is “velocity of B wrt C”. Oddly enough, these don’t sum. But any time “x is x”, it applies.
Regarding syllogisms, let’s not even go there, because they’re flawed (especially when talking about existence).
I would say that the rules of mathematics are invented. However certain consequences of the rules are discovered. The relationship between mathematics and the real world is a mixture of discovery and invention. Generally the rules of mathematics were invented because there is a homomorphism between the mathematical concepts and the real world. So that as a consequence that mathematically 5+4=9 and 9/3=3 and the homomorphism between this and physical objects lets me know that if I have 5 apples and want to give my three friends three each apples I will need to buy 4 more apples. In terms of 4+5=9 being universal. It is true so long as we design a mathematical system that matches the behavior of discrete objects. Any good matching mathematical system will give the same result. We certainly could create mathematical systems that didn’t work this way they just wouldn’t be useful for those purposes.
This isomorphism leads to combinations of invention and discovery with the world. In one direction with new physical discoveries we may invent mathematical systems that are isomorphic with that particular physical reality. In the other direction once we have an homomorphism between the physical and mathematical system, a discovery of the consequences of the mathematical rules can lead to a discovery in the way the physical world works.
Sure, but that doesn’t entail an existence statement regarding abstract mathematical entities (and I wouldn’t exactly say it’s ‘true by definition’; rather, it follows directly from some definitions, and is thus entailed by them).
I didn’t say anything about syllogisms. I gave an example of a sentence whose truth is analytic, and thus knowable a priori, like the truths of mathematics are (a priori, I mean; whether they are synthetic or analytic is of course a matter of some debate, but you, considering them true by definition, seem to hold them to be analytic, so the truth of my example is like those of mathematics on both counts), and for which I think few would hold that it implies the existence of abstract entities.
Agreed, except that depending on how you parse it, “1 + 1 = 2” is true by defintion since S(x) is what defines “+ 1”, and 2 is by definition the successor of 1. Without using the exact terms of number theory, you can’t get much closer to a definition of 2 than this. Perhaps it would be more accurate to say it’s a translation of the definition of 2 (translating from number theory to arithmetic.)
Granted, my criticism doesn’t detract from your point, which was valid.
I said that Russell himself came to consider all of math tautological. I don’t know nearly enough to rule on that myself. But the fact that any sufficiently powerful system is either incomplete or inconsistent implies that it’s definitely not trivial, which is what people usually associate with tautological. IMHO, it’s a hair that’s not necessary to split.
I tend to follow an existentialist line that “existence preceed essense”, for anything that exists in the observable world. But I do question whether that maxim applies to math and logic.
That’s reasonable. But what about the essential math itself? Is there any objective reality to it? Wouldn’t any number of cultures living in completely different worlds come up with essentially the same rules of math and logic? If so, does that give them objective reality, admittedly of a different kind than rocks and sunlight.
Interestingly, it’s rare that physics led to the discovery of new math. (A noted counterexample is probably calculus.) More often, a new math concept is worked out in theory long before we hit physics problems that benefit from it. Examples:
matrix math, which physicists grudgingly adopted to handle thermodynamics
imaginary numbers, used to handle wave equations
and oh crap I know there are more but can’t recall off the top of my head.
Admittedly, that’s beside your point, as either can come first: the tools or the application, but it’s the correspondence that makes the tool useful.
So if we find aliens and they use simple machines does that mean simple machines aren’t inventions and were just floating out there waiting to be discovered? What if they use a lightbulb equivalent? What if they have a story that may as well be Hamlet, but fit to their species?
My argument: the real underlying rules of math are discovered, as they are inherent in the natural universe. I can arrange ten rocks into a rectangle, but not seven rocks. Prime numbers exist, whether or not people have figured out that property.
Created things tend to be individual. Your story, your poem, your painting, your sculpture, all will be very different from mine. But if we’re both given a math homework assignment, we’re going to get the same right answer.*
Somewhere in between is the “art” of proving a mathematical theorem. Here, there is some room for creativity. I might approach the proof differently than you do. There are often more than one pathway through a proof. Also, there are useful mathematical “tools” than can be said to be created, such as Leibniz’ “d” notion for calculus, or Cartesian coordinates and function graphing. These are different from, say, Newton’s “dot” notation, and Polar coordinates; neither can be said to be “right” to the exclusion of the other.
*ETA: or else Mrs. Froiland is gonna make us stay after class and do it all over anew.
Sorry for not being completely on track here, or if I’m just off base in general. I just felt compelled to blabber a bit on certain points raised about creative works, since it’s a field I’m heavily invested in.
I’m not sure I necessarily agree. Creative works are also influenced by everything around us, and what has happened before. You wouldn’t have Throbbing Gristle without Russolo, and you woulnd’t have Russolo without his fondness for machinery, wich you wouldn’t have without the industrial revolution and so on. Lets say I’m writing a story. My story is a product of circumstance. Without those circumstances my story would not be identical to my story. Who knows, if your circumstances are excactly as mine maybe you’d be writing my story and vice versa.
Of course the more rigid a framework, the more identical the results.
And again, the answer may be identical, but how we arrived at that conclusion, the process, might be different. Or maybe my answer is wrong, since I suck at math(I really do).
I don’t think the distinction between invention and discovery is that clear in the first place. To invent is to discover a way.
Concerning math, I think invention and discovery kind of collapse. Take the princple that n+1>n. Do we discover this or invent it? We discover it in the sense that every time we check, either genuinely or “pedagogically” assuming it might not be true, we find it validated. But we invent it in the sense that every time we check on it, we determine what we mean by “+” and “>” and so on.
Given this determination, the discovery follows inevitably. If you think the determination is trivial, you might think the discovery is the significant thing, leading you to say math is discovered, not invented. But:
Once you get past undergraduate math, what the activity of math often consists in is determining how the meaning of previously studied symbols can be extended in new contexts. (And mathematicians often don’t settle on just one extension or another, but explore the implications of all kinds extensions.)
The determination of the meaning of n+1>n isn’t as trivial as it may seem. As human beings, it seems our rational structure very very strongly suggests one particular way to extend the meaning of these symbols when we’re presented with values for ‘n’ we haven’t encountered before. But it would be just as easy for a different rational structure to exist, which extends the symbols differently. For example one which, once you hit the google mark (or whatever), feels like taking away one is the “same activity,” as ±ing had been at every previous step. For this rational structure, it will not always be true that n+1>1. Sure, for this rational structure, it turns out + doesn’t mean the same thing it does for us–but my point here is that determining that meaning (the meaning of +) isn’t in itself such a trivial thing. We bring ourselves actively to every step in that determination. It begins to seem as though we’re discovering, not so much what the world is like out there, but what we ourselves our like. What our rational structure consists in. But we’re “discovering” it by determining it. By inventing it, in other words, though the invention might be quite subconscious. In any case, this is what I mean when I say invention and discovery kind of collapse into one when it comes to math (and many other things)
But I only need to make a small modification to the argument. Every detail of the Cologne Cathedral was designed and held in the mind of somebody. After all, nobody claims that large pieces of architecture are built by one person: it’s a team effort. The Mandelbrot Set, OTOH, still has vast expanses unperceived by anybody. So the set itself cannot have been invented.
Right. But we’re discussing the set itself. And it’s there - it’s real - it’s just embedded in the mathematics itself. It’s not in our minds and it’s not in the computer printouts or posters. (The Russell/Plato half of the OP demonstrates that this isn’t mere mysticism.)
No idea, and I don’t think it matters.
I’m unfamiliar with what you’re talking about, but I don’t think it matters. You can create logical systems - systems that follow a set of rules - that don’t necessarily have any link with the natural world. They subsist on their own. Different assumptions might produce conflicting results: so what? I wouldn’t expect otherwise.
This problem exists even if math is invented… I think (not sure). But frankly I’m not sure there is any causal interaction between the Platonic mathematical world and the empirical one. It’s just that humans armed with knowledge of math can simulate reality. Why this particular logical exercise is so useful is indeed puzzling, but again it would remain so even if math was invented.
I’m puzzled, since I would think that those who reside in flatland or in “4D + time world” would both use linear algebra, understandable and re-discovered in each land.
Frylock: But does math have an existence outside of human thought? Plato and Russell say “Yes”. Math is just an example of a universal, and given Russell’s example in the OP, it’s hard to say that Universal’s don’t subsist. And yet I would have guessed that all that was mystical clap-trap a year ago.
That presupposes that those unperceived parts of the set already exist, which amounts to assuming your conclusion. Every single pixel of the set is drawn according to the recursive formula, and it has to be calculated first before it can be drawn (be that by computer or human calculator). So it, too, existed first in some ‘mind’ (where mind here refers not necessarily to a conscious one).
The problem is, if math exists independently, some part of it can’t both exist and not exist, or be true and false. If Platonism is true, then there ought to be a matter of fact regarding the truth or falsity of the continuum hypothesis—but that does not appear to be the case.
Why? If math is invented (or, as I propose, if it models the structural relations of either real or imagined objects), then it’s in the mind, already, and we don’t have any dualism problem.
But the question is, how they could arrive at this knowledge if the ‘mathematical’ and the ‘physical’ realms are fully separated—because if that were the case, then the mathematical concepts could not make themselves known in the physical world. And if one could then come to mathematical knowledge independently of the mathematical world, then we could as well strike it out of our ontology, as it’s completely without effect.
Nope, it at least points a way towards the resolution of that puzzle: if math is modeled after the properties of objects, then that it embodies such properties is not mysterious. The question is much harder under the assumption of Platonism: here, there’s no a priori reason that the things existing in the mathematical world have any sort of relation to those in the physical world at all.
They can certainly come up with the same formulas as we do, but in their world, what we call ‘the formula for the volume of a sphere’ would not be their formula for the volume of a sphere, so they would not necessarily come up with it. An example is, perhaps, that if you live in a world with birds, and horses, you can come up with the concept of Pegasus, and if you don’t, you probably wouldn’t—but still, you could. Truths about mathematics are like truths about such imaginations: there’s a sense in which ‘Pegasus has wings’ is true, and a sense in which ‘Pegasus is a dog’ is false, but the existence of these truths does not imply the existence (in whatever form) of Pegasus.
Now, mathematics is more constrained than unfettered imagination (mostly by requirements of consistency), so in a sense there’s less freedom to come up with new things, and a greater likelihood to ‘re-invent’ existing concepts, but that doesn’t give them any more of an independent existence than Pegasus has.
Half Man Half Wit: I read your comments with interest. Returning to it today, I think I see where you’re going. I’m not sure whether or not you’ve necessarily excluded neo-Platonicism, though you’ve shown some problems with it.
I had some point-by-point responses drawn up, but I’d first like to ask whether you have a reference that you’re drawing from. I’d be interested in reading the more detailed discussion, if it exists.
A general thought on Ontology. What we believe to be the physical world gets stranger and stranger the further down we go. At the level of quantum theory, everything is only describable using mathematical equations, ordinary language fails us. Same for Cosmology.
I have a floating belief that although we have to accept a sturdy physical universe, the ultimate reality is to do with information and information exchange between things made of such information- an eternal recursiveness.
This woolly concept is one way that I deal with such questions- it collapses the dualism of Descartes as well as explaining some of the questions above.
If physical matter and universals are both comprised of information, then we have a potential monist argument that works.
I’m not drawing from anything specifically, just some motley collection of things I’ve read over the years; most of it’s probably covered in the Stanford Encyclopedia of Philosophy article on the philosophy of mathematics. One other thing I’ve drawn from (mainly confidence that indispensability arguments don’t have the force they’re alleged to) is Hartry Field’s ‘Science without Numbers’, though I don’t really consider myself a nominalist (I’m probably most closely aligned to some variant of structuralism or another, but I wouldn’t really consider my views on the issue anywhere near settled).
I think (though I am not aware of the argument having been developed anywhere) a plausible way of talking about mathematical truths even regarding structures that don’t exist is by following essentially Quine’s strategy in his seminal ‘On What There Is’, where he shows how to deal with assertions about nonexistent entities which we nevertheless regard as having a definite truth value—such as in the case of ‘pegasus has wings’, which, as I said above, is true in a way ‘pegasus is a dog’ is not, even though there is nothing in the world to which ‘pegasus’ refers and that hence could make these assertions true (or false).
So, the study of macroeconomics, especially in the last 40 years or so, has made use of lots of fancy math. The hope is that the fancy math will allow us to make better predictions on what effect certain actions will have on the broader economy, and will help us identify knobs we can turn and levers we can pull on our economic machine that will that make it run better (e.g. employ more people, manage inflation expectations, increase output, etc.). Obviously a macroeconomy is a very complex thing, involving the subjective desires and fears of countless economic actors, combined with the objective constraints of the world we’re operating in. It may be so complex that we never get a good grasp of it, and we’ll only be able to describe it imprecisely and manipulate it crudely. But there’s a school of thought that believes that if we can precisely define the microeconomic foundations of everything that makes up the macroeconomy, then we can sorta just add them all up. Of course, the models that attempt to do this have done a laughably bad job of describing the current macroeconomy and proposing solutions, but those folks haven’t given up.
If we were able to formulate the Grand Unified Theory of Economics, such that we could predict the precise result of a particular policy change or economic variable, and gained full control over the economic knobs and levers, that would be an incredible breakthrough. I can imagine such a theory, but does that mean it’s out there waiting to be discovered? And if it is, is the language of mathematics even capable of describing it? If we can’t describe some phenomenon with math, does the fault lie with us, or with math? If it exists, but math isn’t up to the task of assisting our feeble ape brains in understanding it, isn’t the formulation of that “language of understanding” a creation?