Thread: Do You Think Mathematics Is Natural Or Man-Made?

Originally Posted by Oneiro

What I actually bet her was that I could prove that "one plus one equals one" (spoken, not written). She probably fell for it because I used such basic units, and there was no money involved. She tells me she has used it to great effect with her colleagues. I've told her to call it "The Oneiro Conundrum". Heh.

Yeah, you seem to have a very naive maths professor... and also one apparently lacking in the power of logic and the relevant knowledge. If she lets you make up your own meaning for "1 + 1 = 1" rather than the mutually understood meaning in that context (which is the basis of how language works) then how could she possibly have won; you could have claimed that the sentence corresponds to 'if I jump in the air I will fall back down again' and demonstrated that.

You are hitting on some important issues though. The first is that something has meaning when there is an isomorphism (a one-to-one mapping which preserves structural relationships) to something else. Arithmetic on whole numbers has meaning by virtue of its isomorphism to various agreed kinds of entity (discrete is the key word) and the way they behave with each other. It's worth noting that your non-standard system isomorphic to the world of 'coalescent forms' (incidentally where did you come across this idea? It is touched upon in the awesome book 'Godel, Escher, Bach' where the author talks about raindrops rather than blu-tack) is still totally consistent, although compared to standard arithmetic is extremely trivial (PhilosopherStoned; actually I'd say it is isomorphic to a modular arithmetic, namely modulus 1, if you interpret the symbols correctly; hence also to the trivial group (e, +)). The second issue worth mentioning which should interest you is that the whole numbers were already pinned down formally and unambiguously in the 19th century by Peano in a small number of axioms. This was done via a 'successor function': the number 0 was said to exist, and then you were allowed to create its successor S(0), and then that's successor S(S(0)), then S(S(S(0))) etcetera. The axiom relevant to you is 'for no number is S(n) = 0', which stops the kind of 'looping back' to previous numbers that occurs in your system.

Originally Posted by PhilosopherStoned

Thank you! I can't begin to tell you how many times I need to try to explain this. One operates by proof. The other operates by empiricism (either falsification or confirmation depending on who you ask). End of story.

I don't think agree with this at all. Science operates by logical proof just as much as maths, and maths is based upon empiricism just as much as science. The difference is quantitative, not qualitative: maths tends to use a very small set of observations about more general entities and the logical deductions make up most of the work, whereas science tends to be based on a larger set of empirical data which makes up most of the work, is about more specific entities, and has shorter logical deductions.

A clear example for science is the 'proof' of natural selection as laid out by Ernst Mayr: he provided five observations, three inferences, and the conclusion that natural selection must occur. Natural selection makes a good example because much more of the work than usual (certainly for biology) was in the inference rather than the observation; normally the inferences are tacit. However they are still there. Also, although natural selection concerns a relatively specific class of objects, that is, organisms, it is still about some general, conceptual class of 'thing' and not specific instances.

With mathematics the class of thing is often wider; for example, arithmetic concerns all things that we conceptualise as distinct. It is still however based on observation; for example, the observation that if we combine a collection of objects with a second, and then with a third, there will be the same number as if we'd combined the second with the third and then the first with that. There is no proof here: it is just an appeal to 'reason', which in my opinion we have only obtained by repeated exposure to such instances and then a generalisation, or induction, based on this pattern. Then of course come the inferences based on these observations to deduce new conclusions about our class of objects, as was the case with science.

Having said all that, a final and relatively distinct thing to add: I think inferences are actually just observations anyway. The contrapositive law, for example, is ubiquitous in its application in both mathematics and science, and is a valid single rule of inference. However, how do we deduce it? How do we deduce that P implies Q implies not Q implies not P? I think we have the same situation. P and Q resemble an induction on the pattern of propositions we have encountered. When we try to ascertain the truth of the above, we substitute in this induction and if it works out we conclude that it's okay. I'm in England implies that it's raining. It's not raining. Then I can't be in England, because if I were it'd be raining, which it isn't. Then it checks out, and by induction it checks out for the entire class (propositions) that P and Q represent.

Originally Posted by Xei

Yeah, you seem to have a very naive maths professor...

Been pointed out.

PhilosopherStoned; actually I'd say it is isomorphic to a modular arithmetic, namely modulus 1, if you interpret the symbols correctly; hence also to the trivial group (e, +).

I had a response to this but accidentally deleted it. The rest of my post is long enough and it's a small point. The trivial group has only one element, this has infinite elements so it can't be the trivial group. You can always identify every element (as is the case mod 1) and get the trivial group but that's pretty useless (you can do it with any set) when there's a natural (but uninteresting) monoid structure on it.

I don't think agree with this at all. Science operates by logical proof just as much as maths, and maths is based upon empiricism just as much as science. The difference is quantitative, not qualitative:

The qualitative difference is where in the process these come in. They can both be thought of as broadly operating in 2 steps.

1. Maybe X is true. Maybe it isn't. It's worth looking at. Hypothesis in science. Conjecture in mathematics.
2. demonstrating that X is or is not true. Theory in science. Theorem in mathematics.

Temporarily disregarding your last point, mathematics uses exclusively logical proof in step 2 and empiricism (and other stuff) in step 1. Conversely, science uses logical proof (and other stuff) in step 1 and uses exclusively empiricism step 2.

A clear example for science is the 'proof' of natural selection as laid out by Ernst Mayr: he provided five observations, three inferences, and the conclusion that natural selection must occur. Natural selection makes a good example because much more of the work than usual (certainly for biology) was in the inference rather than the observation; normally the inferences are tacit.

People always want to drag natural selection into this. For the time being, I have to admit natural selection as the one part of science where by statement fails. Personally, I think it's part of 22nd century mathematics that happened to fall in 19th century biology. Consider demonstrating, without empirical means, beyond all shadow of any doubt, that General Relativity (or any other theory of gravity) is true.

However they are still there. Also, although natural selection concerns a relatively specific class of objects, that is, organisms, it is still about some general, conceptual class of 'thing' and not specific instances.

And here's the key right here. It's not really about organisms at all. As we now know, it's about genes and more generally still, it's about any replicating set of objects, e.g., memes, computer programs, etc. It doesn't really matter. Dawkins does a good job of explaining the beginning of this in The Extended Phenotype.

With mathematics the class of thing is often wider; for example, arithmetic concerns all things that we conceptualise as distinct. It is still however based on observation; for example, the observation that if we combine a collection of objects with a second, and then with a third, there will be the same number as if we'd combined the second with the third and then the first with that. There is no proof here: it is just an appeal to 'reason', which in my opinion we have only obtained by repeated exposure to such instances and then a generalisation, or induction, based on this pattern.

There is absolutely proof here. The commutative property is proven for the natural numbers. They are constructed as a sequence of nested sets each of which contains it's predecessor. It is then proven. I've previously referred you to "Introduction to Algebra" by Peter Cameron on this point. I don't have it (or indeed any of my math books) with me at this point but I believe it's in chapter six. It should be at your school's library. Yes, that proof itself rests on the axioms of set theory and we do take axioms eventually but that's really besides the point. How can we be expected to reason without axioms. The point is that those axioms are always clear and explicit. Contrast this with Duhem-Quine Thesis which confounds the ability of science to take axioms.

As for your last point.

Having said all that, a final and relatively distinct thing to add: I think inferences are actually just observations anyway. The contrapositive law, for example, is ubiquitous in its application in both mathematics and science, and is a valid single rule of inference. However, how do we deduce it? How do we deduce that P implies Q implies not Q implies not P?

Assume

1. that P implies Q .
2. not Q

Suppose P.

P implies Q by assumption but we have not Q by assumption. Contradiction.

Hence not P.

QED.

Is there a problem with that?

Originally Posted by PhilosopherStoned

I had a response to this but accidentally deleted it. The rest of my post is long enough and it's a small point. The trivial group has only one element, this has infinite elements so it can't be the trivial group. You can always identify every element (as is the case mod 1) and get the trivial group but that's pretty useless (you can do it with any set) when there's a natural (but uninteresting) monoid structure on it.

To clarify; we were talking about blu-tack arithmetic?

I'm not sure what you mean by infinite elements. Modular arithmetic always has 'infinite elements' from the (uncommon) perspective that you're working with the integers and the operation is 'do something and then take the remainder after division with n', but this is equivalent to an arithmetic with a finite number of elements where the elements are equivalence classes, and also to the finite group Cn.

I was talking of isomorphisms, not homomorphisms (and the mapping you specified, G → {e}, g → e is a homomorphism).

People always want to drag natural selection into this. For the time being, I have to admit natural selection as the one part of science where by statement fails. Personally, I think it's part of 22nd century mathematics that happened to fall in 19th century biology. Consider demonstrating, without empirical means, beyond all shadow of any doubt, that General Relativity (or any other theory of gravity) is true.

I wasn't dragging anything, it just came to mind because I remember it was science that had been explicitly written at some point in observation/inference form. I don't really want to get bogged down in specifics, as I said this is just a good example; however it still occurs in any other area of science, though usually tacitly.

To give another random example... the discovery that ribosomes are involved in protein synthesis. You put ribosomes in a tube with the necessary substrata and get positive tests for polypeptides. You don't put them in and you don't. Here the majority of the science is in observation, but still, there is very simple inference at play here. The first experiment shows they are sufficient, and the second by the contrapositive law shows they are necessary. This is so simple it isn't stated, but it should be clear that we cannot do science without the facility of logical proof.

I don't understand your general relativity point, it seems contrary to what you were trying to say. Saying that GR is provable without empirical evidence is the exact opposite of what I've said here. GR is just another example of how science uses both empirical evidence and logical proof. I specified biology in my post because this very point occurred to me when writing; theoretical science actually has a very high inference to observation ratio. GR rests on a shedload of careful mathematics and hence inference, but the number of observations is very small in comparison (light bending around stars and atomic clocks are the only two that spring to mind). Indeed it is often remarked how theoretical physics is pretty much indistinguishable from mathematics.

And here's the key right here. It's not really about organisms at all. As we now know, it's about genes and more generally still, it's about any replicating set of objects, e.g., memes, computer programs, etc. It doesn't really matter. Dawkins does a good job of explaining the beginning of this in The Extended Phenotype.

I don't see what this is supposed to be key to. Okay, it's more general, and hence it's more like mathematics. The difference is still quantitative and not qualitative which is the only point I've been trying to make (aside from my last).

There is absolutely proof here. The commutative property is proven for the natural numbers. They are constructed as a sequence of nested sets each of which contains it's predecessor. It is then proven. I've previously referred you to "Introduction to Algebra" by Peter Cameron on this point. I don't have it (or indeed any of my math books) with me at this point but I believe it's in chapter six. It should be at your school's library. Yes, that proof itself rests on the axioms of set theory and we do take axioms eventually but that's really besides the point. How can we be expected to reason without axioms. The point is that those axioms are always clear and explicit.

The only reason I took associativity (not commutativity) as an example is because it is visceral and easy to talk about. If you want to go further and talk about the set theory foundation (which I already know about) then fine but it's not besides the point, it's exactly the same point. In fact it's a complete distraction because when you look at the set theory proof it will basically rely on the fact that you can construct a set in any order at some point, which is the same assertion that you can collect distinct objects in any order, and hence everything I said still goes through.

Assume

1. that P implies Q .
2. not Q

Suppose P.

P implies Q by assumption but we have not Q by assumption. Contradiction.

Hence not P.

QED.

Is there a problem with that?

But what is the structure of proof by contradiction?

You have some statement A.

After some work you show that A implies B and it is clear that not B.

Then you say 'hence not A'.

So... you've used the contrapositive law.

And again, even if you can further reduce it, my point still stands for any further reduction.

Originally Posted by Xei

To clarify; we were talking about blu-tack arithmetic?

I'm not sure what you mean by infinite elements. Modular arithmetic always has 'infinite elements' from the (uncommon) perspective that you're working with the integers and the operation is 'do something and then take the remainder after division with n', but this is equivalent to an arithmetic with a finite number of elements where the elements are equivalence classes, and also to the finite group Cn.

Let's stick with the usual definition of modular addition where the elements are cosets of an ideal of the integers and we only look at the additive group structure. Then there's a finite amount of elements unless we take the trivial ideal in which case we get the integers back. So we get the trivial group by taking cosets of the ideal generated by 1 which is the whole ring and the only coset it has is the ring itself. Then we have a group with 1 element which is isomorphic to any other group with 1 element. Agreed?

Now with this "blu-tac" arithmetic, we need to make some sense of it. Take the set of all collections of lumps of clay and impose the equivalence relation on them that we will identify two collections as being the same if they have the same amount of lumps of clay in them. So we have {0 lumps of clay, 1 lump of clay, 2, lumps of clay, ... }. Define addition on this set by mashing together all the lumps of clay in the operands to get one lump of clay. It has the following table

+ 0 1 2 3 ...
0 0 1 1 1
1 1 1 1 1
2 1 1 1 1
... ...

You'll see that it's a monoid: It has 0 for the additive identity, is clearly associative and we clearly have closure. But at the minimum, we have to have two discrete elements to even pretend that we're effectively modeling it: zero lumps of clay and one lump of clay. So the trivial group is right out. There is only one group (C2, Z/2Z, etc.) with two elements and that's out too because we would have to have 1 + 1 = 0 which is not the case here. In fact Cn is out for all n unless we can add 1 lump of clay to n - 1 lumps of clay and end up with 0 lumps of clay.

I was talking of isomorphisms, not homomorphisms (and the mapping you specified, G → {e}, g → e is a homomorphism).

Yes, we're confused. I didn't specify a mapping Now I have. EDIT: Or rather an explicit structure which determines what mappings are possible.

I'm going to have to ruminate over the other (more important) aspect of this discussion.

Okay cool beans, I was considering positive integers and not 0 lumps of clay, and then the isomorphism was from Z1 to {e} via 0 to {clay lump} (both under 'addition'). I haven't actually studied monoids yet, I guess because they're not quite so useful... the higher concepts I know of such as fields are built of groups at least.

The position I'm trying to convey at the moment is one I've only adopted fairly recently actually. For most of my years as a philosopher I've stuck to mathematical platonism and the idea that humans have a grasp of some higher realm, but I suppose only by default. I had a few revelations and found that upon scrutiny the idea of platonism falls down pretty quickly. If you've ever read anything by Roger Penrose you'll know he likes to argue for it, but I've been reading another of his books lately and now all of his mistakes seem blindingly obvious. What I've been left with seems like a very natural and powerful philosophy which makes many of the traditional problems dissolve... it's a work in progress but for epistemological purposes it's basically the statement that the only things we 'know' are patterns of observations; the only thing the human brain can do is identify common instances in scenes and assign symbols to them.

Actually, I just realized that I messed up. The operation table I showed doesn't have 0 as an identity. That would have to be:

+ 0 1 2 3
0 0 1 2 3
1 1 1 1 1
2 2 1 1 1
3 3 1 1 1

So we would have to define addition as mash all the lumps together if at least two lumps are given and leave them alone otherwise. That makes it a monoid. This is getting silly.

But yeah, Saying that something is a monoid allows you to say almost nothing about it so they're pretty useless. That inverse axiom that makes it a group certainly pulls its weight.

Still contemplating the important stuff.

Regarding math and science, I would maintain that there is a meaningful categorical distinction to be drawn between the two, but disagree that it lies in whether each is founded in empirical observation. As I've expressed elsewhere recently, I would agree that it seems that they both rely at some foundational level on empirical observation. Both math and science are processes by which we draw certain inferences, so the key to distinguishing between the two is to look at how these processes differ, not in where their assumptions lie.

With math we have a set of axioms -- wherever they may have come from -- and we are working out what follows from the axioms. We may have derived these axioms empirically (or we may not have--that's not important for my present purpose), but we're not doing math until we're deducing from those axioms.

With science we have a set of observations, and we're trying to reach the axioms. Our belief that it is possible to draw inductive inferences based on specific observations may be based in non-scientific assumptions (once again, that's not important right now), but we're not doing science until we're making inductive inferences about uncertain axioms. (It is perhaps important to note that induction is a necessary but not sufficient criterion for scientific inference; the demarcation problem in philosophy of science is concerned with determining what distinguishes scientific induction from non-scientific induction.)

If you view the important difference between the two processes as being the direction of inference they are attempting, then the distinction couldn't be more categorical. The two are of course inextricably linked -- it takes math to do good science for example -- but any given step is a manifestation of one process, the other process, or neither; but not a combination of both.

That's a very good point. I was really arguing that there is not really any fundamental epistemological difference between the two, but as processes I think they might be dichotomised in such a way.

Although, returning to the example of natural selection; is it not the case that we have the observations, and the trick is to figure out how to use inference to make a conclusion? Is this not more like mathematics than science, in your scheme?

It's trickier than it looks. I think associating the conclusions of science with axioms is questionable... I think that is tacitly relying on the idea that an axiom should be simple, and as in science we often (although it is also important to note, far from always) are engaged in reductionism, down the hierarchy of simplicity, then the conclusions should be associated with axioms. I question this... an axiom is not necessarily something simple, it is just something that we take as known. Surely in science the things we take as known are the empirical data, and these are our axioms? Then we use inference to try and deduce some general theory. This sounds analogous to the mathematical case.

Using the term "axiom" when referring to scientific inference was perhaps a little misleading. I used it only because I wanted to be consistent in my terminology so that I could highlight the fact that I view science and math as being sort of like two opposite lanes on the same road from specifics (observations, proofs, or whatever) to generalities (axioms, laws, or whatever). I don't mean to exactly equate the idea of a scientific law with that of a mathematical axiom, only to point out that they are conceptually similar, in the sense that they refer each to generalities which are thought to hold over all cases in a given class.

For the natural selection example, if we're starting from observations (e.g., geological or genetic facts) and working our way to laws (e.g., gradual evolution by natural selection), I would classify this firmly as a scientific endeavor rather than a mathematical one.

I would agree that the observations are what we take as known when doing science, and so while I see what you mean about these therefore being similar to axioms, that is not quite what I meant it when I invoked the term. Rather, I would say that observations are more analogous to proofs than to axioms (in the sense that observations follow from scientific laws similar to how proofs follow from mathematical axioms), and that the scientific laws that we are attempting to make inductive inferences about are more analogous to axioms than proofs (because, like axioms, we "accept" them even though they are never rendered logically necessary). It's an admittedly imperfect analogy, but again, the emphasis is on the journey from general to specific and back again.

I would say it is a man-made system designed to measure a natural phenomena.

Originally Posted by PhilosopherStoned

Now with this "blu-tac" arithmetic..

Please.. it's the "Oneiro Conundrum"..

Originally Posted by PhilosopherStoned

we need to make some sense of it.

Originally Posted by PhilosopherStoned

Take the set of all collections of lumps of clay and impose the equivalence relation on them that we will identify two collections as being the same if they have the same amount of lumps of clay in them. So we have {0 lumps of clay, 1 lump of clay, 2, lumps of clay, ... }. Define addition on this set by mashing together all the lumps of clay in the operands to get one lump of clay. It has the following table

+ 0 1 2 3 ...
0 0 1 1 1
1 1 1 1 1
2 1 1 1 1

I love it! Ha! Wait 'til I show her!

BTW Guys.. don't fall into the trap of thinking that academics are necessarily blessed with nous. You can trick anyone, if you set up the situation properly.

So.. let me get this right. In order to understand any sum, one must first know the proper definition of the symbols in said sum.. as in "1+1=1" whose symbols can represent different things depending on the case? Bit like semantics in language.

Anyway.. good stuff lads.. keep it up. ;o)

Oh please don't show her that one. Show here the next one I posted where I didn't screw up.

Okay Philosopher.. I'll be kind.. ha! ;o)

The Mathematic System was Discovered by humans, not created: It was inspired on the perfect unwritten equations on which Physical life is based and built up of.
Just take a look at ferns, lotusflowers, Cells, butterflies, patterns on jaguars..etc etc. All based on perfect Geometry. Check out "the Fibernachi sequence" to learn more.
We, Humans, DID however create the numeric system and Algebra; a language to communicate this form of Intelligence and to apply it to every day challenges in Physical Life.

What does the fibonacci sequence prove? What do ferns, flowers, cells, etc. prove other than that humans tend to see familiar patterns in plants that resemble certain logarithmic sequences, due to the fact that if you repeat a similar process over and over again (growing leaves at certain intervals dependant on size and shape for instance) a recognizable pattern is bound to emerge? No fern perfectly fits the particular logarithmic sequence that most closely approximates it. What this proves is that we tend to apply what is familiar to us (our math) to the world that we perceive.

At least numbers are natural. c.f. the natural numbers

You need to use a quantity to describe the size of a rock. Its weight. Its speed. Its height. The force acting on it.
So quantities are natural.
The rock's velocity is related to its acceleration. This is natural. To calculate with this you need MATH.

Nature does math all the time. Planets orbit according to Kepler's laws, which use MATH. Frogs leap because of interactions between molecules, which use MATH. Everything in the world runs on math. Even after the end of the world, if everything is just mixed up, the powdered fragments of Earth will still follow the MATHEMATIC laws of physics as they crash into DV's poor server, initiating a short circuit, the heat output of it coming from MATH.

That shows mathematics arises naturally as a description, or model, of what we see the universe doing, but what does it mean to say that mathematics is natural?

The brightness of a torch dims in proportion to the inverse of the square of the distance of separation, but that does not mean that b = k*d^2 is inherently real. Rather, it's a model that's a product of generalising the mass flow of photons (think about this; after 1 distance the light takes up 1 square, after 2 distances it takes up 4 squares, after 3 distances 9 squares; draw it if you want), so really the maths is just a result of what happens when lots of things spread out in a straight line from a point, certainly not vice-versa.

As a mathematician, this question -off course- enormously intrigues me. I'm on neither side, because I tend to shift from believing it is all a man-made construction to believing it is some devine underlying structure and back and forth. So I will not disrupt this discussion which is an excellent exercise in philosophy and critical thinking. But for who wants to read more (and didn't already) I'd like to point to the following text which elaborate on the subject:

Indispensability Arguments in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)

Consequences of Incompleteness Theorems. Unsolvability. Creativity in Mathematics. Size of Proofs. Related Results. By K.Podnieks

[0704.0646] The Mathematical Universe

BTW. I didn't read every post, but I think Brouwer wasn't mentioned yet. It is really a brilliant and intriguing man, whose philosophy tends to be overlooked.

Intuitionism and constructivism

Intuitionism was proposed by Brouwer as a philosophy of mathematics without foundations. Whereas Kant had sort to ground arithmetic in the experience of time and geometry in the experience of space, Brouwer attempted to account for all of mathematics in terms of the intuition (conscious experience) of time. Intuitionism clashed with classical mathematics in so far as Brouwer held that there are no truths beyond experience, and hence that the law of the excluded middle could not be applied to all mathematical statements (in particular infinitary parts of mathematics are indeterminate with respect to some properties). The successor of intuitionism, namely constructivism, has abandoned Kantian metaphysics and epistemology but still maintains that some mathematical statements, namely those for which no proof has been constructed, have no truth value.