Wednesday, May 3rd, 2006
This post comes follows from these steps. My good friend Jean-Marc proposed this Cartesian enclosure:
Mathematics is the only sure path that leads to heaven. Remember that God created the integers and everything else is the work of man.
Note that I take this in the psirit of fun in which it was offered and pretend no expertise. Here’s my response
Can one prove the existence of integers? Let’s take the positive integer 2. Two what?
Here’s Jean-Marc’s response
The question of existence or inexistence of integers (or mathematical objects in general) is purely philosophical. Hence, it is hard to debate. Nonetheless, you must recall the 3 main dogmas that are at the core of the Foundations of Mathematics: Platonism, formalisms, and constructivism.
The Platonist holds that mathematical objects are real, immutable, and independent of the human mind. So, according to those folks, mathematics is already out there waiting to be discovered. The formalist, in contrast, claims that mathematics is a compilation of definitions, axioms, and theorems that are not about anything real. The constructivist’s view is quite different from the previous two as he views mathematics as a collection of ideas that can be obtained via finite constructions. So, the Cantor Set (that you and I talked about recently) would be seen as useless and a complete waste of time by the constructivist.
I hope this answers your questions on existence or inexistence of mathematical objects. With respect to your comment “Let’s take the positive integer 2. Two what?” I am not sure I know what you mean here. Are you talking about the numeral 2 or the number 2? Either way, your answer is in the preceding paragraph.
I want to end this email on a note of curiosity. How do String and Quantum disprove Leopold Kronecker’s view that “God created the integers”? My curiosity will only be satisfied if your response is not philosophical. I would want to see an argument that is deductive in nature.
Unfortunately, the original question was philosophical re: Kronecker, who perhaps should not have ascribed integers to God, unless he could prove that tensor theory was also a divine creation. My response is, I believe, non-philosophical and asks for some demonstration of God’s hand in the creation of integers as a matter of play. Troubling scientific waters, I would claim.
Therefore, the question “2 what?” asserts that it’s the oranges that are real not their relation to one another and asserts further that integers can only be the work of man.
As to the platonist, formalist, constructivist triplet, consider the Kronecker Delta. Do we know what “idea” the symbol refers to? We know the formalism. And can build build 3index objects. Isn’t it the elements that matter?