| Subject: Axiomatically true mathematical systems |
Author:
Damoclese
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Date Posted: 03/24/04 7:00am
The field of mathematics is much larger than reality itself. By this I mean that there are mathematical models out there that are logically consistent within mathematics, but don't exist in reality.
Non-euclidian geometry is a case in point. There are many models of geometry that are self-consistent, but seem to have nothing in reality of which to base themselves in. Because they are axiomatically self-consistent, they are said to be mathematically true.
Now, if one wishes to "reject" one of these models, the only way it can be done is to look in reality and see what can be explained or what can't be explained by them. If reality doesn't support them, that doesn't mean a particular premise will without fail, necessarily be wrong. In fact, it's already known to be right, at least mathematically.
I think this fact would be prudent to remember, because there are ways in which logical arguments fail besides within the premises themselves. While one can be mathematically consistent, it doesn't mean the model in question will be useful, relevant, or helpful. And really, it's because we don't see the thing in reality that the argument fails.
The questions I'm inclined to ask Wade in light of this fact are: Do you see a beginningless task in reality conducted by humans? Yes? No? If no, then there is at least one assumption that is imaginary, though mathematically possible.
Do we have any examples of an infinite past in reality that we can point squarely at so that we have some basis for what to expect from an infinite past? Yes? No? If not, then we have another premise that is imaginary in nature, and doesn't necessarily have any particular consequences a priori in reality.
Do these objections "disprove" the validity of the argument? Of course not in a strictly logical sense, but then the entire argument is relegated to a place where it might be possible, but it certainly doesn't seem to be the case based upon what we see in reality.
Therefore, I am of the opinion though while paradoxes are often great for mental masturbation as it were, if they rest on things that don't seem to exist in reality then we are forced to place them in the mathematical waste bin of things that could be true but surely don't seem to be.
And that, is the real reason to reject the Tristam Shandy argument, along with Zeno's argument. It's simply the difference between mathematics and science.
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