Hi Marc --it's interesting to wonder about "what it would be like" to directly perceive mathematics -- but we also have to acknowledge when we ask the question, what are the philosophical assumptions we're smuggling along. For instance, the human brain is not capable of direct perception of tables, either.
What raises a flag for me in your question is the following apparent dichotomy: 1) "The human brain is not capable of direct perception of mathematical entities" 2) "We could imagine some super-intelligence that possessed this ability . . ."
It seems what you're encouraging us to do is this: think about of what it's like when we see a table, and then say to ourselves something like the following sentence: "It would be like that, but with *math*". But what makes us think we can imagine this situation coherently? Light from a table excites our photoreceptors in a well- understood way - how could an equation do that?
I have always thought it strange how McGinn and others eagerly apply "cognitive closure" to some of the very areas where we have made recent amazing progress in understanding! In the case of math, what exactly is it that motivates your intuition that there might be something more that we're missing? And is it something that would not apply trivially to any other thing (i.e. - I can look at a rock on the ground, and say to myself, "There's something else about this rock that I'm not sensing - but I could imagine a superintelligence who could perceive what I'm missing." My ability to say this sentence to myself doesn't demonstrate anything interesting about the rock.)
Best regards Pete On Jan 27, 2006, at 1:08 AM, Marc Geddes wrote:
Open question here: What is mathematics? ;)A series of intuitions I've been having have started to suggest to me that mathematics may not at all be what we think it is!The idea of 'cognitive closure' (Colin McGinn) looms large here. The human brain is not capable of direct perception of mathematical entities. We cannot 'see' mathematics directly in the same way we 'see' a table for instance. This of course may not say much about the nature of mathematics, but more about the limitations of the human brain. Suppose then, that some variant of platonism is true and mathematical entities exist 'out there' and there is *in principle* a modality ( a method of sensory perception like hearing, sight, taste) for direct perception of mathematics. We could imagine some super-intelligence that possessed this ability to directly perceive mathematics. What would this super- intelligence 'see' ?Perhaps there's something of enormous importance about the nature of mathematics that every one has over-looked so far, something that would be obvious to the super-intelligence with the mathematical modality? Are we all over-looking some incredible truths here? Again, McGinn's idea of cognitive closure is that the human brain may be 'cognitively closed' with respect to some truths because the physical equipment is not up to the job - like the way a dog cannot learn Chinese for instance.For one thing: Are platonic mathematical entities really static and timeless like platonist philosophers say? What if platonic mathematical entities can 'change state' somehow ? What if they're dynamic? And what if the *movement* of platonic mathematics entities *are* Qualia (conscious experiences). Are there any mathematical truths which may be time indexed (time dependent)? Or are all mathematical truths really fixed?The Platonists says that mathematics under-pins reality, but what is the *relationship* between mathematical, mental (teleological) and physical properties? How do mental (teleological/volitional) and physical properties *emerge* from mathematics? That's what every one is missing and what has not been explained.So... think on my questions. Is there something HUGE we all missing as regards the nature of mathematics? Is mathematics really what you think it is? ;)--"Till shade is gone, till water is gone, into the shadow with teeth bared, screaming defiance with the last breath, to spit in Sightblinder's eye on the last day"
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