Science’s Relationship to Logic and Mathematics

03Oct12

Whether and to what extent logic and mathematics are sciences is currently a hot topic of discussion in philosophy.  Much of this discussion has to do with the debate over whether or not science can provide methods and insights for studying and understanding all areas of human endeavor.  (As an often overlooked corollary this includes guiding change in those areas using scientific tools and methods.)  Those who deny that science can provide methods and insights in all areas raise logic and mathematics as exemplar areas in which science fails.

What makes logic and mathematics subject to the use of scientific methods is the fact that they are areas of human endeavor, or practice.  Science may be applied to these areas because they arise out of the material and contain material aspects.  The material is matter.  Matter is that which exists in time and space.  Matter can be reflected in thought, but exists outside of thought and in that sense matter exists objectively.  The word objectively is not being used here in the sense of something that can be understood outside of a context, but rather that regardless of the meaning an object of matter may have in a given context it does exist in time and space, it exist objectively.  Activity or endeavor is a material thing because it involves material objects and processes.  When you are carrying out an activity or endeavor you are activating or intervening on material objects.  Even thought is material in that it takes place in a material vessel, the brain, as the physical basis of the mind.  (The mind being a product of the brain in interaction with its environment and recursively with itself.)

Science studies objects, structures and processes in time and space.  Science attempts to verify the hypothesis generated from its studies in order to be able to label the results of that study as knowledge.  Often we can then use that knowledge to change the world.  Potentially all things in time and space are able to be analyzed, understood and changed via science.  “Potentially” because for instance as with string theory, if strings exist then our present tools do not allow us to see them.  Logic and mathematics as pursuits first arose out of the activity of humans changing the world that exists in time and space.  The material processes from which they arose and many of the processes we know they affect are not beyond the reach of our tools and thus we can apply science to and within logic and mathematics.

Many of the truths of mathematics are “a priori” (they exist prior to human practice, or experience).  This fact does not put mathematics outside of science.  Mathematics of the “a priori” more often than not is an abstract reflection of how matter interacts with itself.  And just as physical science studies the truth of particles that existed before humans, so science can be applied to mathematics that exist prior to humans insofar as the mathematics is an abstract reflection of the way the material interacts.  Given this nature of mathematics it should be no surprise that mathematicians who often use axioms based upon the material in a fundamental sense, find that discoveries once thought to have no material counterpart are later found to have a connection with some area of the sciences, technology or the social sciences.

Other truths in mathematics are deemed analytic (true by definition of the terms used) like 2 + 2 = 4.  In this case isn’t it material experience that has taught us what 2 things are and that adding 2 things makes 4 of them?  Still other mathematical truths are so because they are based upon systems of self-enclosed axioms.  However, where mathematics meets life outside of mathematics, and that is many, many places, the truth of its statements can and must be contextually verified in human practice.  Study of the material in human practice and verification of relevant hypothesis is the core of scientific method.

The same arguments made about mathematics apply to logic, whose axioms in many of it branches often begins with observations about the way things relate to one another in the material world of human practice outside of logic per se.

-Tanner Phillips, @philosophyofsci

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