I used to be convinced that mathematical objects are in some sense real. That numbers have an existence independent of humans knowledge of them. Recently, in a Wittgensteinian turn, I’ve come to the realization that the question “are numbers real” is a bad one.
The heart of the problem is that the question is ambiguous. Math is a set of models that attempt to describe phenomena we observe in reality. Those models are obviously invented. We didn’t discover them carved into a rock somewhere. The phenomena in the world they are describing are obviously discovered. So the answer to the (badly phrased) question is both. “Numbers” are a feature of a model humans invented to describe a physical phenomena. The physical phenomena that numbers model is something humans discovered in the world. The only way we have access to those things is through the intermediary that is our models. But this introduces a new problem.
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Logic, Mathematics, Metaphysics
Or, You Can’t Know What You Don’t Know You Can’t Know.
This is an interesting result from modal logic that I will try to sketch here. The upshot of the result, depending on which side of a divide you fall into, is either that there are some truths that are logically impossible to know, or that every truth is already known by someone.
The dividing line in this case is whether you are a realist or anti-realist. The realists posit that there is an external reality that has certain definite properties. The anti-realist deny that such an external reality exists (or, in some cases, that we can have access to it). I’ll get more into this distinction after I sketch the proof. If you find logic boring, feel free to skip the proof and scroll to the end for a brief discussion on the implications of this result.
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Logic, Philosophy
XKCD posted a logic program titled Blue Eyes: The Hardest Logic Puzzle in the World. The problem got posted to reddit, and of course a large argument erupted in the comments thread about how the question doesn’t make sense, the solution doesn’t make sense / doesn’t work / is flawed / etc. etc. etc.
Rather than futilely attempt to make myself heard above the din, I’m writing this post which explains what the solution is, how you get to the conclusion, why it is in fact correct, and why the guru’s statement is necessary.
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Evolution, Logic
The paradox of the heap, also known as the Sorities Paradox (from the Greek word for heap), is a paradox revolving around the problem of vagueness.
In its classical formulation, the paradox is expressed as follows:
One grain of sand is not a heap.
If one grain of sand is not a heap, adding one grain of sand will not make it a heap.
So two grains of sand are not a heap.
So three grains of sand do not make a heap.
…
X grains of sand do not make a heap.
Therefore, 10,000 grains of sand do not make a heap.
The form of this argument boils down to:
X grains of sand are not a heap.
If X grains of sand are not a heap, adding 1 grain of sand will not make it a heap.
(Some arbitrary large number of grains of sand) do not make a heap.
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Computer Science, Evolution, Musings, Philosophy, Vagueness
In one of those funny coincidences where it seems the popular mind is pregnant with an idea, an article was recently published to the programming section on reddit that mentions the busy beaver function. The article is titled “Who Can Name The Bigger Number?” and is essentially about trying to name very large integers.
The article is fairly long, but also quite interesting. The associated thread on reddit is also interesting, but is topped by a very similar discussion on the XKCD blog about naming large numbers.
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Logic, Mathematics
Computers are a wonderful invention, capable of a profound variety of actions. But there are limits to what a computer can do, and today I wish to talk about one of them. Specifically, I wish is discuss the Busy Beaver problem.
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Computer Science, Logic