How Does A Materialist Account for Logic?
From Logic
You probably already know how this complaint goes:
"How can you account for axioms in a materialistic universe? What part of your brain are axioms located in? Can you actually point to some neurons and say 'these are what the axioms really are'? Also, since the axioms of math are carried around in people's heads, are there really millions of little axioms of math running around? Finally, how come you also call an axiom written on the page the axiom' and the axiom in your head 'the axiom'? After all, paper isn't a bunch of neurons, and you are a materialist after all..."
Let's take this apart, piece by piece:
How do you account for the 'laws of logic' in a materialistic universe?
This question contains a false presumption. And to reveal it, I ask the following: 'The' laws of logic? Which set of laws? For which logic? First-order logic, first-order predicate logic, second-order predicate logic, modal logic, fuzzy logic? Which one? Logic is not a monolithic entity, and there is no one set of 'laws' for all of logic. Some logical systems do not require axioms at all. The set of axioms for the sentential, or propositional, logic is {} - the empty set.
"The point is that there can be no axioms in propositional logic, the most basic of all modern logics (there are other formulations that do have axioms, though): everything in predicate logic is definitional. And how does one argue with a definition? My point is: the answer to the question "why doesn't everyone accept the axioms of logic?" is that it can be the case that there's nothing to accept. Literally."- Gregory Lopez.
The Basic Metaphysical Requirements for any Logical System
Now that we have done away with the blunders attached to that misunderstanding, let's explain the basic metaphysics required for the creation of an a priori system - the existence of sentient brains. The basic axioms of existence, identity and consciousness - the so called laws of reason (which are prior to any logical system and NOT part of logic itself), are necessary elements of reason; to reason one must first exist, and exist as something. These axioms are therefore implicitly inescapable - an explicit awareness of these axioms is another matter.
We can express this truth thusly:
To exist is to exist as something. And to be aware of this, is to be conscious. To update Descartes, we might say:
"I exist, therefore I think"
These axioms of reason are necessary truths, given the existence of consciousness. They are defended through retortion. But they are not a part of logic, they exist prior to the formation (or learning) of any set of logical rules. Other rules, such as the other laws of classical logic, can also be gleaned a priori, all of them flow from the axiom of identity (i.e. classical logic, a system of tautologies, can be traced back to the axiom of identity). The specifics of which rules we create do not matter here; what matters is as long as we have sentient brains, we will have the basis for the creation of any a priori system.
To 'explain' the human invention of logical systems as somehow the work of God is to explain precisely nothing at all. No one who ever asserts this putative 'solution' ever defines this 'god' in a noncontradictory way, nor do they ever explain how this 'god' provides the foundation for logic. If they were to undertake an examination of their assertion they'd soon recognize that referencing the supernatural as foundation for a natural system leads to an infinitely greater problem than working out the metaphysics for logic, to whit: supernatural references are broken terms. Invoking "god" to explain anything is the very abdication of reason and insight.
What part of your brain are axioms (or abstractions) located in?
The cerebral cortex, frontal lobes. http://www.waiting.com/brainanatomy.html#anchor2587568
"Also, since the axioms of math are carried around in people's heads, are there really billions of little axioms of math running around?"
Billions of representations of the same axioms. Billions of sentient brains coming to the same, necessary, analytic, unavoidable, a priori conclusion, just as billions of different bits of falling matter all conform to the same phenomenon of nature that we can summarize in one law: the law of gravity.
If you fail to find it puzzling how 'different pieces of matter' can all conform to the same law of gravity, then you ought to re-examine your supposed puzzlement over axioms. The process is similar. Billions of sentient brains encountering the same, singular reality - the unavoidable basic metaphysics of our universe. If you are looking for a missing 'constant' for the materialistic account, it is this: the universe. You've misplaced the universe. One universe with a basic set of unavoidable, inescapable metaphysics. One universe imprinting itself onto phylogenetically similar sentient beings, who are able to draw the same abstractions from the same stimuli, based on the same rules...
Axioms, are abstractions that exist in a brain. The reason we see the 'same axiom' in different brains is because the same idea can be gleaned, analytically, a priori, by similar brains in the same exact universe. The same idea can be represented in multiple copies - the same firing of neurons in my brain as someone else's (more or less), which then become emergent phenomenon such as "abstract concepts" to our consciousnesses.
Finally, how come you also call an axiom written on the page the axiom' and the axiom in your head 'the axiom'? After all, paper isn't a bunch of neurons, and you are a materialist after all...
Ah, but you forget something else: Abstract entities written on a page have no meaning in and of themselves. They are interactive phenomena - a sentient brain is required to interpret them and provide them with 'meaning'. Thus, when we say that the number "eighteen" is written on a page, what materialists are really saying is that this sensory input'18' through some social convention (some rule), yields the same firing of neurons in my brain as someone else's (more or less), which then become emergent phenomenon such as "abstract concepts" to our consciousnesses.
A word written on a page and the same word spoken and traveling as a wave through the air are not 'the same matter'. However, when I read the word, and when I hear the word, my brain eventually interprets them the same way, producing similar electrochemical responses with enough fidelity that slightly different brains can reproduce the same abstraction, based on the same rules.
Of course, the mapping itself is completely arbitrary. Our written alphabet needn't be what it is, and we could choose totally different symbols to represent the same thing as the spoken word.
Common Responses
The Identity of Indiscernibles is usually formulated as follows: if, for every property F, object x has F if and only if object y has F, then x is identical to y. Or in the notation of symbolic logic:
∀F(Fx ↔ Fy) → x=y.
This formula can be used to demonstrate that if x shares the same properties of y, then x and y are the same entity.
The argument continues:
A material entity cannot be in more than one spatio-temporal location at the same time.
Response: This claim is built on intellectual dishonesty, for it fails to consider that abstractions are tokens or representations - formed in neurons by the same set of rules. However, leaving this aside, the claim fails even if we accept this bit of dishonest, as quantum physics tells us that there is no contradiction in having the same material entity in more than one spatio-temporal location:
http://www.fizyka.umk.pl/~jkob/physnews/node30.html
For more in the principle of Identity of indiscernibles and Liebnitz' transposition of the principle, the law of Indiscernability of Identicals, see here:
http://plato.stanford.edu/entries/identity-indiscernible/#Rec
