A couple of posts ago I asked if real numbers exist (like pi).

It really doesn’t matter. and I’ve come back around (one of these mental oscillations…) to the conclusion brought to my attention on the NKS blog.

Here is the key statement:

Mathematics is a symbolic language — you can argue that none of its elements “exist” in physical reality, yet they can be used to communicate information about things which are real.

I found another statement to this effect in the classic “What is Mathematics?” by Courant & Robbins, revised by Stewart.

Through the ages mathematicians have considered their objects such as numbers, points, etc., as substantial things in themselves. Since these entities had always defied attempts at an adequate description, it slowly dawned on the mathematicians of the the nineteenth century that the question of the meaning of these objects as substantial things does not make sense within mathematics, if at all. The only relevant assertions concerning them do no refer to substantial reality; they state only the interrelations between mathematically “undefined objects” and the rules governing operations with them.” What points, lines, numbers “actually” are cannot and need not be discussed in mathematical science. What matters and what corresponds to “verifiable” fact is structure and relationship, that two points determine a line, that numbers combine according to certain rules to form other numbers, etc.

I often forget that the abstraction is not the thing. The metaphor is not the thing. The symbol is not the thing. Mathematics never makes assertions that it is the thing. It is an abstraction – a description of relationships devoid of many of specific objects’ and environments’ properties. This abstraction (and simplification) is required to make progress. If mathematicians were to create theory that was specific to every situation, object, and environment, the world would run out of shelf space for storing all the math books and we’d gain nothing over flat out recording keeping. In a sense, mathematical abstraction is a wonderfully useful compression of information. The application of mathematics to a specific situation is the decompression of the abstraction.

This abstraction is so useful because it lets us focus on key relations and make progress on understanding despite our lack of complete knowledge of specific objects, environments, and situations.

This abstraction is also dangerous and/or limiting. Not all situations in the universe are able to be described by a simplified mathematical theory. In fact, a surprising number of very simple phenomena (theoretical, biological, physical, financial, etc.) are not mathematically compressible. That is, a purely mathematical theory will not be sufficient for understanding in many situations.

What a relief!

Some mathematicians already experienced this relief from needing to describe the universe in some ultimate truth. That’s far too big for any discipline to bear.

Not all mathematicians and a majority of economists, business people, vcs, media, and lay people do not recognize this limitation of mathematical theory (heck, and many other theories!). In the US (perhaps elsewhere), business models (pro formas), stock indexes, indicators, projections, forecasts, formulas dominate our thinking on very complex phenomena. We’ve explored this issue many times on this blog.

Understanding the universe we experience requires a combination of theories. Math can sometimes point the way and get us going, keep us focused, or help us communicate.

Whether the real numbers exist doesn’t really matter. The real numbers are useful for moving us forward on some problems in the real world. Pi, as a compression of a really long number and challenging concept in geometric forms, is useful in helping us make wheels, explore space, and so much more. That is what makes math great, even if it isn’t objective, ultimate truth.

on December 27, 2008 at 10:24 pm |LeoI agree ´cause if some people say that pi is not constructable with straightedge and compass then, a circle is also impossible to construct with a compass…By the way, to me pi is constructable….and can prove it

on May 20, 2010 at 7:44 am |David Wheeleri AM a mathematician, and i am not so sure. i am familiar with the various ways of describing the real numbers, and all of them have to do with ensuring that we can study “the continuum”. the question is, is continuity actually a correct model of the world we live in? i agree it’s a useful model; but newtonian physics was a useful model until we began to learn more about the very small, and the very large, and has now been by and large supplanted by relativity and quantum theory.

i think this is a deeper question than most people realize, and quite crucial to the foundation of mathematics. perhaps one day, existence of the reals will be one possible mathematics, just as allowing or denying the axiom of choice gives rise to different fundamental theories of sets.

if at some level of existence, space/time/”stuff” no longer dispays the properties of continuousness we take for granted on the scale(s) we habitually inhabit, and are no longer valid, we may want to reconsider our notions of how we model space and time.

i dispute the validity of the statement pi is computable. as i understand it, a program written to calculate pi will NEVER terminate. just saying you can get the error of such a calculation as small as you like, may not be good enough. one of the troubles with real numbers is that the vast majority of them aren’t even possible to name. in fact, given any 2 rational numbers, no matter how close together, you cannot even name the real numbers between them.

now, i’m not advocating doing away with all the useful results the real number system (and the calculi it has spawned) has given us. i’m saying that the way(s) we have come up with the real number system are indicative of a lack of genuine understanding into the nature of the world we live in. i fear that we will come to regard them as a problem solved, rather than a mystery to continue to be explored.