Saturday, January 29, 2011

Mathematics: Invented or Discovered?

For my history of math course we were asked to write an essay on the topic "Is math invented or discovered?" This is my response.


The debate regarding the ontology of mathematics is a philosophical quandary that extends deep into our cognitive history, near the emergence of sincere cognizance itself. This fact is hinted at in the division of arguments, in which a significant subset is Platonism. The classic allegory of the cave is often illustrated with a specific chair in the room as an oblique projection of the form that unifies all chairs under the notion of chair-ness, some vague set of qualities that result in an object being classified as a chair. This approach does elicit some glimmer of understanding, but the allegory has a vastly more ornate interpretation with consideration of the forms as abstract mathematics and the shadows as specific instances of those general principles. From this perspective there is some credence in the conjecture that Plato was influenced by the ontology of the Pythagoreans, which held that “Everything is number.” With mathematical forms as eternal and unchanging, a Platonist concludes that mathematics is discovered. 
 
A common reaction to this conclusion is the proposed problem of a priori existence, which cites the contradiction that the nonphysical forms exist without a physical manifestation before attaining representation as chemicals in the brain. If mathematics can only exist as an arrangement of physical objects, and these arrangements can only be produced by consciousness, it is reasonable to conclude that mathematics is invented through intelligent processing of experience. 
 
Despite so much thought on the invention vs. discovery of mathematics, the question is broken--clearly a false dichotomy--which really should have been recognized after all the contradictions started arising. It might be helpful to approach this question with a set theoretic interpretation of language. Let each word be a set comprised of its synonyms, including itself (as a singular element), and its definitions. Considering the word roots A = “invent” and B = “discover,” most modern references will give A intersect B as not null, frequently even giving the subset {invent, discover}. Now the question is if, for the word “mathematics” = M, 
 
(A is a member of M) OR (B is a member of M)

but this is a misrepresentation of the problem since A and B are not mutually exclusive, thus mathematics might be a member of invented, discovered, both, or neither.

Both the failure and success of language are a result of its persistent nebulosity, which enables the vaguely logical cogitation that profoundly influences our consciousness; to this we owe our capacity to experience the wonder of metaphor and the sincere difficulty of attaining certainty. As experienced language users, we know that what constitutes a word is not limited only to definitions and synonyms, that language does not naturally obey logic, which is why non formalized philosophical debate can proceed indefinitely. Accordingly, we can redefine the question and approach from another direction entirely, with definitions that appear mutually exclusive. Consider the two statements that thefreedictionary offers on the page for “discover”:
  • We discover something that existed but was not yet known.
  • We invent something that was not in existence.
Note that the second statement implies something, for if something was not in existence, it must not have been known, so
  • We discover something that existed but was not yet known.
  • We invent something that was not in existence and was not yet known.
which reduces to
  • We discover something that existed.
  • We invent something that was not in existence.
Thus distinguishing between invention and discovery relies entirely on existence; now we must ascertain if there is a difference between existence and non-existence. Consider the following definitions taken from thefreedictionary:
  • Exist: To have actual being; be real.
  • Existence: The fact or state of existing; being.
  • Being: The state or quality of having existence.
  • Real: being or occurring in fact or actuality; having verifiable existence.
  • Actuality: The state or fact of being actual; reality. See Synonyms at existence.
  • Actual: Existing and not merely potential or possible. See Synonyms at real.
  • Fact: Something demonstrated to exist or known to have existed; believed to be true or real.
  • True: Consistent with fact or reality; not false or erroneous. See Synonyms at real.
An attempt to simplify the definition of “exist” by replacing words with their definitions results in nonsense along the lines of
  • Exist: To have the fact of existence; having existence in fact or the fact of having existence; having existence or occurring in fact or the fact of having existence existing; having verifiable existence.
In lieu of a definition that consists of something other than self substantiation, let the definition of existence be the following,
  • Exist/Existence: true.
where “true” is in accordance with the familiar logic construct. Now we have a definition which is very useful for a formal analysis of the problem. If non-existence is not true, then there is no non-existence and everything exists; if non-existence is true, then non-existence must actually be existence by definition, therefore everything exists and everything, including mathematics, is discovered.

But this answer is contrived and not inviolable, because the resolution of the problem in the system of logical analysis, like all, depends entirely on the definitions. We can reach the opposite conclusion by giving an alternate definition, which would clearly result in mathematics classified as invented:
  • Existence: the quality gained by something when it is first represented in a human brain.
Thus, in order to answer a question, the terms must be well defined, which is not the case for this ontological debate.

1 comment:

jman said...

very well written =)