Friday, December 5, 2014

Math and reality

It's hard to imagine science without any kind of math.  It gets even harder as you move down to the more fundamental sciences, such as physics.  So many concepts in physics are mathematical, such as mass, velocity, speed, force, energy, etc.  From physics and upwards to the special sciences, so many laws are stated as mathematical equations.  E=mc^2, F=ma, etc.

Now, the sciences are investigations into reality.  What does it signify when we conclude that math is an inextricable part of science?

We take it for granted that so much of what we observe in the world is describable in terms of quantities and relations between quantities.  But, the world needn't have turned out that way.  The physical universe could have had no discernible pattern, thus resisting the formulation of mathematical laws describing the behavior of objects.

So why is it that math so effectively describes the world?  One possible explanation is simply that math is just as much a part of reality as atoms and molecules are.  Ontology is the study of existence, as well as what exists.  We often think of reality in ontological terms, i.e. giving an inventory of what exists.  However, this isn't all there is to reality.  Reality is not just about listing things that exists.  There is also the structure of reality, i.e. the way in which existing things are related.  Structure matters just as much in understanding reality as ontology does.  For example, just listing a bunch of wooden planks isn't enough to understand why something is a boat.  A pile of wood doesn't get you said boat.  You need something more.  You need to know how those planks are related, i.e. how they're put together.  Likewise a full understanding of reality means acknowledging that reality includes both things and structure.

So, we can say that math is a part of reality in that it is the structure of reality.

We could say the same thing about logic.  Math is to physics as logic is to metaphysics, although perhaps to a lesser degree in the case of metaphysics.  One can think of physics as a way of applying mathematics.  Similarly, one can think of metaphysics as a way of applying logic.  If logic plays a role that is similar to the role that mathematics plays in physics, then one can infer by analogy that logic is also part of the structure of reality.  This shouldn't be surprising, given the close association between mathematics and logic.

One consequence of the view that numbers and logical concepts are just as much a part of reality as tables and planets are is that there is more than one way to gain knowledge about objective reality.  Scientific knowledge is knowledge gained primarily via observation, i.e. through sense perception.  Logic and mathematical knowledge is knowledge gained through what I call "rational insight."  I don't use the word "intuition" because of its connotation with the kind of snap judgment thinking that you read about in Malcolm Gladwell's Blink and Daniel Kahneman's Thinking, Fast and Slow.  Rational insight is the faculty through which we come to know that claims in math and logic are true.  It is distinct from sense perception, in that it does not involve the use of our eyes, ears, etc.

So, if we accept that mathematical and logical entities are part of the physical universe, then it turns out that perhaps philosophy, metaphysics in particular, might tell us just as much about the universe as physics does.

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