Presented to the Socratic Club at CBU by Matthew Yocum on October 25th, 2007
During the nineteenth century the thought of empiricism began to pervade the sciences. The traditional platonic view of mathematics (i.e. numbers existing in and of themselves, and the discovery of said entities) began to lose support. Thus differing philosophies of mathematics arose out of the subsequent collapse trying to prove that mathematics is a valid science. Three main schools of thought were produced. Other schools of thought arose out of these three, but we shall ignore most of those for now. This paper will make a straw man of each view in opposition to my own and attempt to tear them to shreds.
Since the thesis indicated that all of the view’s truthiness will be judged by my own view, we will start there. My incoherent view is as follows: (1) Numbers and mathematical principles do not need any human to think of them in order for them to exist. (2) Numbers and mathematical principles are part of the framework of the universe. (3) Numbers and mathematical principles, while not purely logic in and of themselves, must follow logic when they are stated in an equation. For example:
Two is not equal to four.
Two and three are equal to five.
Seven is greater than zero.
Five is equal to five multiplied by five divided by five.
The statements above, albeit simple, must always be true. 2 can equal 3 as much as I have a circle-square in my pocket. (4) Numbers are contingent upon their relationship to one another, i.e. you cannot have 3 without 2 and so on.
The first school of thought is Logicism. The logicist attempts to reduce mathematics to pure logic. This was done to solve the apparent ontological problems with the platonic view of numbers. This was “proved” by a series of equations, now called into question, that are far too complicated to explain here.
There are questions I have regarding numbers and mathematics as logic. Mathematic equations seem to follow along the same lines as a deductive argument; the conclusion must be true if the premises are true. For example, 2 + 3 must equal 5. 2, 3, and 5 are all symbols, but are the concepts they represent logic? Does listing mathematics and numbers under the realm of logic solve the ontology and empirical problem?
The second school of thought is intuitionalism. Intuitionalists believe that mathematics is a human construction. Numbers and mathematics are a construct of the human mind. They are therefore only limited by the finite human. Because of this, the intuitionalist view holds that there is no such thing as an actual infinite.
I would disagree with this view because numbers are not a human construction; mathematics and numbers seem be woven into the very fabric of the universe. Intuitionalists believe that humans created numbers by observing quantities. For example, if farmer A had 2 cows and farmer B had 3 cows without the farmers having any concept of numbers or mathematics, it would be obvious to the farmers that farmer B has more cows (Figure 1).
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Figure 1.
The farmers would observe this over a period of time and would add linguistic qualities to the numbers themselves. I agree that humans can observe quantities, but human observation alone does not determine or exclude existence. The humans do not need to observe that 3 cows are more than 2 in order for that to be true. Along those lines, it does not necessarily follow that humans have to think of the concept of numbers in order for them to exist.
Formalism is the third school of thought. The school’s though is similar to intuitionalism, but does not believe that numbers are a mental construction. The formalist believes that numbers are purely symbols and symbols are abstract entities. Mathematical concepts are not absolutely true, “mathematics is no more than a game in which symbols are manipulated according to fixed rules” (Horsten, 2007). I do not see how this works. How can the formalist not believe that numbers are a mental construction and then say they are of a theoretical existence?
The last of the schools of thought is fictionalism. The fictionalist holds that there are no mathematical principles or numbers. Numbers and mathematical principles are like fiction stories. Mathematics describes fictional entities. There are problems inherent in the name fictional entities; what entities are fictional entities? (Horsten, 2007).
In the end, this is all naught before my eyes. Life’s most burning question has not been answered: Why is there the same amount of even numbers as there are numbers themselves? That is to say, how can the whole set of even numbers be equal to the whole set of natural numbers if they both are infinite. It seems as if this paper has been reduced to reductio ad absurdum or three lines on top of one another.
This paper was plagiarized mostly from these sources:
Horsten, L. “Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Winter 2007 Edition), Edward N. Zalta (ed.), forthcoming URL = http://plato.stanford.edu/archives/win2007/entries/philosophy-mathematics/
Jech, T. “Set Theory”, The Stanford Encyclopedia of Philosophy (Fall 2002 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2002/entries/set-theory/>.
“Philosophy of mathematics.” Wikipedia, The Free Encyclopedia. 15 Oct 2007, 16:18 UTC. Wikimedia Foundation, Inc. 25 Oct 2007 <http://en.wikipedia.org/w/index.php?title=Philosophy_of_mathematics&oldid=164739097>.
Rafalko, R. J. Logic for an Overcast Tuesday. Belmont: Wadsworth Inc.,1990
