Mathematics and Reality

In the first part of Roger Penrose’s book The Road to Reality (which I have not yet finished) Penrose argues that mathematics has an objective reality. It’s not entirely clear to me what he means by this, but I suspect that he is wrong. He points out that everybody who studies math comes to the same conclusions, and that seriously studying math has the feel of uncovering deeper and deeper truths. He also brings in Mandelbrot’s set, though I’m not sure why since it doesn’t seem to support his thesis.

It is undeniable that math is a deep subject, and that mathematics is an investigation shared by the human race across time. However, this does not mean that math has an objective reality beyond its existence in humanity’s shared knowledge. I think the easiest way to see this is to compare it to newer branch of knowledge, such as cellular automata.

The study of cellular automata is not mathematics. Cellular automata are abstract, but they are not based on numbers or geometry. Cellular automata are nevertheless a complex field of study, as can be seen by the extensive studies of just one instance, Conway’s Game of Life. There are undoubtedly general laws and insights waiting to be found in the study of cellular automatons.

However, I think it is silly to argue that cellular automatons have any sort of objective existence in a Platonic ideal space. The only way to support that is to become a thoroughgoing Platonist and believe that there exists an ideal object for all abstract concepts. So since I don’t see any essential difference between the study of mathematics and the study of cellular automata, I don’t see any reason to believe that mathematics has any objective reality.

There is a related issue, which is why reality is apparently describable in mathematical terms. Of course this is in part probably just an effect of how we observe things: we describe reality in terms of mathematics because those parts of reality which can not be described that way tend not to be described at all.

For example, consider the physics of gases. There are simple equations describing the relationship between the volume, temperature and pressure of an ideal gas. These equations do a good job of describing reality. However, we know that these equations are really just a way of describing the interactions of many many individual molecules. In this case, we are able to reduce the complexity of tracking individual molecules with simple mathematical formulas. Note that the gas laws are perfectly real: they correctly predict and describe the behaviour of gases. However, there is an underlying, more complex, system which produces the results the gas laws describe.

The weather on the Earth is also a complex set of interactions of many many individual molecules. However, in this case we have so far completely failed to describe the weather using mathematical formulas. Our best attempts to describe the weather involve complex models–in fact, a form of cellular automata. So when we say that reality can be described by mathematics, we evidently don’t mean the weather. Are we simply picking and choosing the things which we can describe? Also, when we find a simple mathematical formula in physics, I think we always have to ask: is this reality, or is it a formula which manages to capture underlying complex behaviour?

There is another way to consider the relationship between mathematics and reality, based on the weak anthropic principle. We are complex creatures, and our complexity was formed by evolutionary processes over a long period of time. Evolution is a very flexible technique, but it can only create complex objects when it operates in a relatively static reality. If the laws of physics changed unpredictably, it’s difficult to see how evolution would ever be able to build up complexity. Therefore, since we exist, we can conclude that the reality we inhabit must be relatively static. And mathematics is well suited to a description of many static systems. This does not prove that reality must be describable mathematically, but it suggests that we need not be surprised that it can.


  1. fche said,

    December 31, 2005 @ 6:16 am

    Can you explain why you believe that the existence of an underlying complex system somehow negates the claim that mathematics can describe a phenomenon? That underlying system can be made the target of subsequent mathematical study (as in statistical mechanics for gases).

  2. Ian Lance Taylor said,

    December 31, 2005 @ 1:00 pm

    I see that I didn’t give a good explanation of what I was thinking about. The question is not whether mathematics can describe reality; that is given. The question is whether there is a deep connection between mathematics and reality. Is reality inherently mathematical? When we study mathematics, are we studying something which is somehow tied deeply into reality? Is there something privileged about mathematics, beyond other types of human study? I think Penrose would say yes. I think Kant might say that mathematics is inherent in the way we view reality, though it may not be inherent in reality itself. I’m trying to say that there may not be a deep connection at all, using the physics of gases as an example of how mathematics can successfully describe reality without being deeply connected to it.

  3. fche said,

    January 1, 2006 @ 5:38 pm

    It makes one wonder what nature a “deep connection” between mathematics and reality could possibly be in order to satisfy the inquiry. I am satisfied to call something as “objective” or “real” if it meets some standard of proof, sort of like in the criminal justice system, or in the “consensus theory of reality”. If one seeks a more formal isomorphism between the real and abstract, one would have to define what mappings would be acceptable.

  4. Ian Lance Taylor said,

    January 1, 2006 @ 10:58 pm

    To me, an example of a deep connection would be something like Newton’s universal theory of gravitation, F = -G * (m1 * m2) / r**2. If that were a complete description of reality, not created out of underlying elements, then reality would be inherently mathematical.

    As to the objective reality of mathematics, that has to be interpreted in a Platonic sense; Plato argued that the ideal chair must really exist, because otherwise it would not be possible to speak of chairs as a concept at all. Most people today do not believe that, although I have spoken with people who do. I think Penrose was arguing that mathematics really does exist in the Platonic sense, so that mathematicians are not studying an abtract field of thought, they are studying a real object, or exploring a real universe, accessible through thought.

    As far as I can see, these distinctions can not make any practical difference. It’s all just philosophical fun and games.

  5. Deep Mathematics I: “Typical” Theorems « The Twenty Eighth Line said,

    August 18, 2008 @ 3:18 am

    […] Deep Mathematics I: “Typical” Theorems This is the first of what I hope will be a series of (not chronolgically consecutive) posts discussing what makes “deep” mathematics deep. This is, of course, not a new issue and has been dicussed many times before. I suppose this is more an issue of semantics than philosophy, but this is certainly one place the fact that mathematics is done by human beings rather just being being the “cold and austere” * discipline that it is so often viewed as being becomes apparant. […]

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