This question is really more philosophical than anything else, but does anyone have any suggestions as to where we get our mathematical axioms. Most people would probably say that we get them from what we observe to be true in nature. The only problem with this is that we are placing logic on the same level as intuition. We can't do this as mathematics is, beyond doubt, certain, yet intuition tends to be generally very wrong. So where do we get logic and mathematics?

Brad

I think that we do get our intuitions from nature, and this affects our choice of axioms, and unfortunately this means that mathematics isn't beyond doubt. In the past I've been severely bogged down by this sort of question, in fact I'm reading Kant's Critique of Pure Reason even now to try and help me get a better grip on questions very much like this. In the end, however, there is very little, perhaps nothing, that you can say with absolute certainty.

There's a story by Jorge-Luis Borges (an excellent writer, you should definitely read him), called The Blue Tiger or something like that, in which there are these blue pebbles which defy mathematics, the number of them varies at different times, and each time they are counted there are a different number of them. The story deals with it quite interestingly and intelligently.

The point, to mathematicians, is that the laws of logic and mathematics are not true a priori (which means they are not true to the extent that their not being true is unimaginable). What mathematics does help with is this: If we assume some very basic facts about which nobody can have any serious doubts, such as (a+b)+c=a+(b+c) for integers, then we can deduce things strictly logically (assuming also that logic works as we expect it to). In other words, it clarifies what we should believe about complex issues, depending on what we believe about simple issues, the idea being that we should be more clear about simple issues.

In mathematics, this is a very powerful principle, but it doesn't work for all disciplines, as I've discovered. My favourite example is the political principles of Equality and Inheritance, most people fundamentally believe that (a) Everyone should have an equal chance in life, (b) People should be able to leave their belongings to their children. Unfortunately these are contradictory. What can be deduced from this I leave as an exercise to the reader.

Well, that's my current take on the issue of truth and certainty, sorry it's so pessimistic. It hasn't stopped me loving mathematics however, but it has made me see everything in a different light.

My question is not really about where mathematical axioms come from, but rather why mathematics and other laws of nature actually exist at all. I know this really is a stupid question, because of course no one knows, but I really want to see how others think about this.

How come there is mathematics, and laws of nature? I mean, I always wondered why nature actually behaved in such an orderly and simple manner. Why not at random? Why is it that whenever we add 1+1, we always get 2? (In base 10, of course) This always suggests to me that there must be an all-powerful being out there controlling and setting up these laws, otherwise it would be extremely unlikely that chance has caused this laws of nature to suddenly "pop out" out of nothing! But then, this argument always worries me because, if there are beings controlling our universe, where the heck did they come from? Were they also created by other even more powerful beings? If so, where did THESE in turn come from?

What do you think?

I think there is some confusion in the above regarding distinguishing between mathematics and physics. In particular, it is not suprising that 1+1=2, this is just a definition. It IS suprising that 1 potato + 1 potato = 2 potatoes, this is what is brought out in the cobble story by Borges that Dan refers to.

Logic is true a priori, it is just a construction of definitions, however, that it should be true in the world is another matter, as Tony is pointing out.

Now on the other hand, it is not entirely surprising that maths works, because a good reason for constructing the integers is that they do model things like sheep, cobbles an potatoes accurately.

Kant doesn´t really distinguish these points, I think, because he thinks logic is imposed on the world by the way we look at it; he claims that in the same way that we cannot see things outside of space and time, we cannot see them outside of, say, a causal structure (effect-cause) or a law of the excluded middle (A or not A must be true). Now this can be a useful way of looking at things, but it does seem to suppose unnecessarily that logic is embedded in the world a priori, whilst, as Borges points out, it need not be.

Sean

The problem with logic being true a priori is that if it is based off of nature (it's what we observe), then it could always end up being wrong (so long as we define true to be that which coheres with what is seen). I suppose that a good example that this is true is the statement that if something is in room A, then it can't be in a different room. But, quantum mechanics mandates that it can be in both places at once. This fact, once thought to be logic, is no longer that widely used, especially in physics.

But why have mathematics, what in my interpretation is based upon logic, never seen to be wrong. Not one thing in it is wrong to desribe the universe either (so long as the universe possesses the properties of the mathematics we are talking about). This is mainly aimed at number theory, since it is the root of all other mathematics. Simply put, why is number theory (1 book+1 book=2 books, etc.) always right, and not based off of an incorrect observation? Is it just that this observation is so simple that it stands little chance of being disproved? Are all the observations in maths (mainly the axioms of algebra, the building blocks for all else) just this simple?(obviously geometries must be excluded as they deal with thing based on a very risky set of axioms)

Brad

Dear Brad and others,
a lovely thread of conversation. Thank-you! A deep and yet accessible introduction to this topic may be found in Douglas Hoffstadter's Godel, Escher and Bach.

He helpfully points out the corrolary to Godel's theorem: we can never be sure if our axoims are consistent. If they are, they certainly won't be complete (i.e. enough to eliminate all doubt about what is and is not true).

The axioms we use are conventions. They've been chosen to give a *model* of the world: there's a utility (and hence funding) to the research mathematics in this mathematical universe. The utility being that *so far* they haven't been falsified when applied across the realms from abstraction to the real world. It is this interpretation which makes mathematics 'applied' and in turn determines which of a choice of axioms is adopted by the mainstream.

But it isn't the case that there is only one set of mathematical axioms. A straightforward example being the parallel postulate. In plane (Euclidean) geometry parallel lines never meet: angles of a triangle sum to 180deg. This need not be the case. On a sphere, parallel lines meet in exacltly two places (the poles) and angles of a triangle sum to more than 180deg(draw one on a balloon to *test* this!). The two versions of mathematics offered here are both valid, but inconsistent. They lead to different models of the world which have different uses.

Many other, more subtle choices face mathematicians over *which* mathematics they wish to work in.

A lovely example of this is a cicumstance in which a conjecture is proven to be unproveable (see earlier comments re Godel). One cannot say with certainty if it is true or false. So a mathematician can choose either without fear of contradiction (else it *could* be proved - by contradiction). This leads to two mathematical universes splitting off based on the axioms that conjecture X is either true or false. What is extraordinary in this circumstance is that this branching in the tree of mathematical truth can partially rejoin should both assumptions lead to the same theorem. I understand this exact situation has occured in respect of certain results in number theory, but I'm ignorant of what. A muppet's guide is invited...

Cheers, Pete Capewell

I think that the statement:

mathematics "clarifies what we should believe about complex issues, depending on what we believe about simple issues"

summarises applied mathematics perfectly.

Pure mathematics doesn't actually say anything at all about the real world - it is a totally self-sustained field. Therefore it doesn't require any assumptions or beliefs to begin with (in theory at least - in practice it doesn't work like this because the theory is never totally rigorous from the start.) Instead, it sets up basic definitions and then deduces theorems about these, from the original definitions. However the objects you find in pure maths (some random examples: numbers, equations, functions, operations and geometry) are not chosen by accident. They are chosen because they can be used to form easy parallels with the real world, and hence help the other main area: applied maths. Also it is easy to make observations about familiar objects.

It would technically be possible to develop a pure mathematical system without numbers, or functions, or geometry (and as long as it was self-consistent it would be fine) but this would be less useful as a tool for analysing the world.

Yours,

Michael

Tony, OK, I misunderstood your question, what you are asking is actually much more interesting than what I thought you were asking. Is there some explanation for why we see order in nature? There might be an anthropological principle argument that explains it to some extent. The anthropological principle basically says "The reason that we observe behaviour ABC rather than XYZ is that if XYZ were true, then humans couldn't exist, so we wouldn't be here to see it." For instance, the "reason" that the universe is 3D rather than some other dimension is that in higher dimensions, orbits are unstable (assuming gravity is inverse cubed for 4D, etc.) so there couldn't be planets for us to live on and that 2D organisms are impossible (well, maybe they are possible, but they're unlikely). It's a debatable principle, but there is some small amount of credence in it. Back to the question. Maybe if there were no order, the sort of structures that can appreciate order (i.e. us) cannot form, because order-appreciating structures must be orderly themselves. It's far-fetched, but somewhat compelling (I like it anyway). The other nice thing about the anthropological principle is that it bypasses the need for god, or other controlling beings.

An alternative explanation is that it might be the case that all very simple systems have an order at a higher level, for instance there is a school of thought that a sufficiently large random "Game of Life" board will create intelligent beings after a sufficiently large number of generations. It's pure speculation, but it might be that the universe is something like a Game of Life board (we can't see it as that though, because we are part of the board ourselves), and that a small group of very simple rules generates all the rules that physics has worked out, including rules like 1 potato + 1 potato = 2 potatoes. I read something about this recently in New Scientist, if you go to their web page and search for "random universe" or something like that, you might find the article.

I don't know neccesarily that the anthropologic principle allows for a God not to exist. My take on it (in its primary application) is that the quantum mechanics needed for A to occur will occur so that the consequences of A occur. But, there still must be a reason for A to occur. In the case above, it's quantum mechanics. In some cases it could be God; this case doesn't neccesarily exist in our universe, the formation of our universe could just as easily be explained by a lack of causation outside of our universe. But the anthropologic principle, for all its merit, still must have a reason for why behind it (and the cause can't be the effect).

Brad

Brad - there is a slight confusion in the way you are relating QM and the law of the excluded middle (which say either A or not A must be true). This DOES hold in QM, the thing is the basic object is not where is the object, but what is the wavefunction. An object being at a given point means the wavefunction is a delta function at that point, and therefore cannot be somewhere else. If the wavefunction is not a delta function at a point that the concept of where the object is makes no sense, so you cannot apply the law of the excluded middle to the whereabouts of the object because the whereabouts of the object is not defined.

So the law of excluded middle in QM says the wavefunction cannot be two different functions at once. Its not that the object is in various places at once, but that the concept of absolute locations is meaningless.

It is true that the anthropic principle does not supply a cause. However, if you combine the anthropic principle with a random generation of universes then you are not lacking anything.

But I do think the anthropic principle does somehow sweep the problem under the rug, so, for that matter, does God. My own opinion is that what is needed are a series of revolutionary advances in physics that will change our perspective as much as QM and GR have done (easier said than done!!) and that this new advance may let us try to approach this problems, but at the moment I do not think we have even the concepts or the language to do it. I do not think the notions of space, time and causality will survive in their present form.
Sean

Dan - the Game of Life is deceptively simple. As you may know, if we situatate the rules of the game in a parameter space of cellular automata, where the space is of all the possible rules for a 2D automata, then the Game of Life is located right on a phase transition between rules in which all cells die and rules in which you get complete randomness. This makes the system particularly interesting, because many features of like happen at phase transitions (for example the lipid cell walls are quasi-crystals at a liquid/solid phase transition) at we would expect anyway that life requires just the right amount of complication.

Sean

I think that what the anthropological principle shows is that it isn't inconceivable that there is an explanation of the order we see without resorting to God, the principle might not be the correct explanation however. The final answer to "why does something exist rather than nothing" however remains intractable of course. Sean, yes, the Game of Life is "just right", so perhaps if we combine the idea of the Game of Life with the anthropological principle, we get the world as we know it? Who knows? Fun ideas anyway.

Firstly, I'll echo Pete Capewell's reccommendation of "Godel, Escher, Bach: an Eternal Golden Braid", which every sentient being should read.

The anthropic principle, when you get down to it, is "our being here implies an ordered universe". It says nothing about why the Universe is ordered. The "why" of physical laws is a deep and unresolvable question.

As for the "why" of mathematical axioms, that is another thing entirely. Number theory is not the foundation of all mathematics, for my money that's set theory. You can define everything, including numbers, starting from a foundation of set theoretic axioms and the empty set. There is a school of mathematical philosophy which holds that only those pieces of mathematics that can be rigorously proved from these foundations are valid.

Brad, you talk about logic and QM. There is research in logic (a tiny part of it was mine) where you start with beliefs that are not certain - "I'm 70\% sure that the particle is in room A" - and see what deductions you can make from them. In rigorous proof and argument, however, it is true to say that everything rests on the validity of your axioms. How do we know these are true? We don't. We choose them because they seem to be true, and often because their results are nice. There is a set theoretic axiom, the "Axiom of Choice", which is frequently cited. If anyone were ever to somehow prove it false based on "better" axioms, huge amounts of mathematics would fall down.

In the end, we don't care about the "why" of axioms as much as we care about where they lead us.

Marcus, are you talking about fuzzy logic above? If so, I've been looking for a non-commercial application of fuzzy logic, can you help? I just read a book about it which gave the distinct impression that it was an engineering trick rather than a deep theoretical structure.

It depends on what you mean by "fuzzy logic". There are at least a couple of distinct ways you can look at it. The first is where the fuzziness lies only in the implications - "X almost always implies Y". I don't know much about that one. Another (reasoning under uncertainty, which I hope I know something about since I got a PhD in the area) involves assigning truth values to statements of propositional or predicate logic. In traditional logic, a statement R(c) (which we might interpret as "it is raining in Cambridge") can have truth value 1 (true) or 0 (false). When we deal in uncertain reasoning, we give it a truth value in the range [0,1], with 0 meaning definitely false, 1 meaning definitely true and values in between denoting degrees of belief in the truth of the statement. I might give the statement a truth value of 0.8, since the wind is blowing from the south and it rained here in Surrey earlier. You, on the other hand, can just look out of the window and assign a 0 or 1. As you can see, the truth function depends on the individual. You can formulate rules for manipulating these truth values. Much more is known about propositional uncertain reasoning (where you only have statements such as R - say "it is raining", and can manipulate them using AND, OR, IMPLIES and so on) than for predicate reasoning (where you can say R(c), or R(x) for all x). The former is implemented, with some simple rules for manipulation, in some "expert systems".

The research that I did was in predicate uncertain reasoning. The application is about as noncommercial as you can get. What you do is formulate some assumptions about how a rational being's belief functions would change under certain circumstances - adding new knowledge. The belief function is a map from a certain set of core statements in the logical language to the interval [0,1]. When you make the various assumptions, you see what mathematical structure is imposed on the truth functions. There is a deep theoretical and philosophical structure here. You have to justify your assumptions - for example, "in the absence of any knowledge about either, you must assign the same truth value to statements P and Q". When you have too many assumptions, however, it often turns out they are contradictory, or give rise to a trivial belief function. Once you have put the mathematical structure together, you then have to try it out to see if its "decisions" ring true. The problem with any commercial application for this is that for any nontrivial knowledge base the calculations needed for the theoretically "best" functions are hideously time consuming.

I could go on at even greater length, but I'll spare you.

If you are interested in uncertain reasoning, and you have a good grounding in logic and probability, a good text is "The Uncertain Reasoner's Companion: A Mathematical Perspective" by J. B. Paris, C.U.P. 1994 - or you could mail me and have me ramble incoherently about my research...

I dont know if anyone else has mentioned this but there was one case that I can think of where the intution of a mathematican was more astounding than you could basically imagine. In the 20th century, at the beginning in fact there was a young uneducated Indian man who would prove to be the most fabulous and extraordinary mathematican of last century. He was called Ramanujan, I sure alot of you have heard or know about him and what was the most extraordinary thing about him was that most of the new discoveries that he made was due to intuition and nothing else. A formula that would have baffled the best for years would suddenly come to him in the night and he would jot it down and show it to the other mathematicans who would be the ones that would have to struggle to find a proof for it! and in most cases the proof was extremely difficult. Now remember most ordinary mathematicans have to work for years to discover new maths and this guy did it in his sleep and had it recorded within an hour!.