To properly ask that question, you must first define 1 and 2 (and + and =, for that matter).
To properly make that statement you must first define ‘properly’, ‘question’, ‘first’, and ‘define’. That’s one of the reasons I say math is discovered not invented. We invent the definitions, we discover what they are defining.
Mangetout, the process to make strawberry jam always existed based on the definitions of strawberry, jam, and every other little bit of the process. You invent the symbolically described process, but in some unlikely circumstances strawberries could fall off the bush into a little puddle of a hot spring and turn into jam, with no one there to do the inventing.
Imagine we traveled to another planet and met an intelligent species that had evolved completely independent of humanity. Despite this separate existence, I would expect we would find there are things we have in common: we’d both identify the same elements, we’d both have the same speed for light, and we’d both use the same mathematical formula to calculate the volume of a sphere - these are universals that exist independently of who finds them. They are therefore discovered not invented.
I am by fiat going to declare my opinion that mathematics is a discovered reality rather than an invented one.
And then I am going to duck out and avoid the hurled rotten tomatos.
You are correct, the OP is trippy. But the core arguments come from Bertrand Russell, who was a diamond-hard logician, at least in his philosophical writings.
I’m puzzled by this post. I was arguing above that mathematics pre-exists before it is put in communicable form. Just like rocks and plants. I used to believe that rocks and plants existed before somebody put them in communicable form, but that math and poetry were inventions.
I don’t know enough either, so I’ll free-associate. Russell’s 1912 work mentioned in the OP was an introductory text. In the larger discussion, he covers philosophers like Hume and Kant who had a different epistemological take. He thinks their problem was that they focused on nouns rather than verbs. So when Plato talks about the Universal chair as opposed to a particular chair and then claims that the former is the real one, many such as myself become extremely dubious. But when you consider verbs or rather relations, then the concept of the Universal makes more sense, to me at least.
Half Man Wit: Thanks for your posts. I frankly need to think them over. This post of mine doesn’t address the key issues.
To clarify, I’m not claiming that Mandelbrot sets represent anything in nature. I’m only claiming they are very detailed. (Yes, I’ve set aside the main part of your argument - again I need to think this over.)
I don’t think it’s that simple. I don’t doubt the existence of matter, energy or thought. If there’s something that pre-exists in the world and it is described, we can call that a discovery. Sure, there will be an aspect of artistic expression involved.
In terms of the philosophical literature, we are exploring a priori reasoning.
Nice treatment. Playing Devil’s Advocate, I suppose fiction has its own internal logic as well: see TVtropes.com for examples. And biologists possess a concept of convergent evolution whereby cactuses in the Sahara and Southwestern US have entirely different lineages yet display similar physical traits. So while your argument is certainly indicative (and heck, I for one find it persuasive) it may have some holes that need patching.
Many thanks to the replies in this thread. I have much to ponder.
In Z[sub]2[/sub], the finite field with two elements, 1 + 1 = 0. We use 1 + 1 = 2 more often because it’s well modelled by the things around us, but as I said above, there are cases that conform better to the other equation (and to all other possibilities). The fundamental fact is the existence of (real or conceivable) objects that model the relation; the relation itself has no independent existence.
The formula we use to calculate the volume of a sphere is dictated by the particular kind of space we live in, it’s not an ‘extra’ fact. Beings living in other, nonflat spaces will use other formulae. That doesn’t mean they couldn’t have come up with the formula we use, as well: they could imagine a different kind of space, and investigate its properties. There’s just no need to require an independent existence of the formula, and doing so creates more problems than it solves (such as the problem of interaction all dualism entails).
FTR, that would be Roger Penrose (2004). The copyright at Amazon is 2007, but I assume that’s a revised edition.
I would disagree. I feel there is a direct one-to-one correspondence of discovered concepts like chemistry or physics or mathematics. There will be different symbols but the symbols will define the same thing. If we explain what we mean when we talk about hydrogen or gravity or right angles, the aliens will say “Oh, you’re talking about f7e## or 88b&s or )k;35p). We know about that.”
But if you took something like Hamlet, you could explain it to the aliens and translate it into their language. But they wouldn’t say, “Oh, we’ve got the same play. Only we call it gT4s*$.”
As for convergent evolution, I intentionally chose aliens to eliminate arguments that all people (and all other life on Earth) originated from a single source. So the parallel discovery of the same mathematical concept on two different continents might just reflect the effects of shared DNA.
I disagree. I think that these aliens would live in the same universe that we live in and would therefore discover the same principles that we’ve discovered. So their spheres would be identical to our spheres and their formulae would be identical to our formulae.
This doesn’t seem like a really useful definition of ‘exist’ - in fact, I can’t think of any use for it than to beg the question in a debate such as this one.
Of course 1+1=2 could be used as the definition of 2. For example I could define “phlarg to the jth goba equals flomat”, but does that mean that “phlarg to the jth goba equals flomat” is a universal truth?
What I’m saying is, 1+1 and 2 are just two different ways of saying the same thing. There is no new information there.
The process of addition, and techniques for actually calculating it, are useful to a human being however, as it may not be self-evident to us that 31313 + 81921 = 113234.
What’s interesting is if you follow this train of thought to what it is that computers do.
When a computer simulates a storm, say, we assume that in some sense what is happening in the computer maps to what is happening in a storm. Indeed, some people even might like to think a kind of storm is happening in the computer’s memory.
I am suggesting that this is not so, and what a computer is simulating is a set of processes a human would follow. That is, the way a human would take known facts about a storm, and manipulate that information to work out new facts. These processes need have no relationship at all to what is happening in a storm.</possible hijack>
The fundamentals of the universe (cause and effect, etc) and the fundamentals of mathematics all share some properties, but this is because mathematics began as a way of describing the universe, or aspects of the world. If the language (of maths) isn’t in accord with what it’s trying to describe, it will be inaccurate.
And so perhaps in another hypothetical universe, where perhaps what we would think of as an effect precedes cause, their mathematics would be more complicated in it’s fundamentals; Even in the definitions of simple operations like of arithmetic.
Seems you could apply the concept of discovery to almost anything that has to meet a criteria, e.g. Edison didn’t invent the lightbulb, he just searched the space of all possible ways to make a lightbulb using 19th century techology and found one design that worked well enough for what he was trying to do.
Math is what it is. How it became known… is irrelevant.
No, if you examine the universe as it exists, you will find things like substances and motion and even numbers. And by studying these things, you can discover things like chemistry and physics and mathematics.
But you won’t find a lightbulb in nature. To get a lightbulb you have to start by studying nature, then figure out natural laws, and then apply those natural laws in ways that nature did not. That’s invention.
But then, the formula depends on a contingent property of our universe—in which case it does not have any independent existence. Only if you held that independently of the particulars of the universe, this formula were necessary, then you would claim it to exist in itself. Otherwise, it’s a property of space.
Besides, there exist regions in our universe where the flat-space sphere volume formula does not apply, and while it’s unlikely, it’s perhaps not impossible that intelligent life might develop there.
Then why isn’t 1 + 1 always 2? In order to make 1 + 1 come out right, you need to fix a certain axiom system of which the desired outcome is a theorem; there’s no a priori reason that the Peano axioms for natural numbers do a better job of that than the axioms of the two element field, for instance.
I did not claim that any area of maths can be applied to any real life situation.
What I’m saying is that maths is a set of tools for manipulating information. And that it is not surprising that some maths has practical value because it is essentially augmenting our own thinking processes.
Now, the mathematics of a two-element field are a set of axioms by which we can take facts about a two element system and derive novel facts about a two element system. You’re asking me why this doesn’t work on a system we have no reason to suppose is a two element system. I would think the answer to this is obvious.
OTOH if we apply this mathematics to a system that is two-element (e.g. a toggle switch), it works.
I think it’s a bit of a semantic question. Normally semantic questions are boring (because they can be easily resolved, just stipulate a set of terms and their definitions.) In this case, I think it’s more interesting because it shows a weakness in English.
Mathematics is both invention and discovery. Or rather, it’s neither alone.
Invention requires creativity. Discovery, at its simplest, is just tripping over something.
But what mathematicians do is a subtle mixture of the two. This is also true of what physicists do. Yes, there’s a distinction between math and physics, but in both cases, we invent a framework for analysis of some underlying pattern that already exists. There’s just no good word for it, and we fall back on the two mentioned above.
With math, the distinction is even narrower than for physics. For physics, we can say that the physical phenomenon described by a theory existed before the theory did. For the math, it’s not so clear: the mathematical truth existed before the mathematical statement was ever made. (Then again, some may say that the physical law existed before the theory behind it was stated, so maybe they’re not so different in this regard.)
Of course, math and physics are very different in that the latter is empirical, but that’s beside my point here. It’s generally taken as a given that there is no possible universe where mathematical truth would be different.
Big fan of Russell, btw.
But you did say that ‘1 + 1’ and ‘2’ are just two ways of saying the same thing; I merely pointed out that that’s not the case. It’s only the case when talking about the appropriate sort of objects; but then, why should the truth of ‘1 + 1 = 2’ be an independent and further fact, i.e. exist apart from the objects that embody the relation it expresses?
Consider a sentence like ‘all bachelors are unmarried’. It’s certainly true (given the right definition of the included terms, which in the mathematics case are provided by the axioms), but there’s nothing that needs to ‘exist’ to make it true—indeed, even if everybody were married, the sentence would still express a truth. I need not invoke any sort of abstract Platonic realm in which there exists an ideal bachelor in order to make it come out true, and neither do I need to do so for mathematical statements.