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1RaymondGrey
Back in autumn, I was watching my grandson throw a Frisbee to his dog, Ralph, who was making perfect catches, and it occurred to me that Ralph was making excellent analog calculations of the Frisbee’s flight path in order to position himself for each catch. I tried an experiment: I got a slightly larger and different colored Frisbee from the trunk of my car, one which Ralph had never seen before, and gave it to my grandson to try. Ralph made the transition from one Frisbee to the other almost seamlessly. Apparently, he was able to recognize the general value of his analog calculations and apply them to the new object’s flight path.
Some would assert that mathematics has an independent existence, that it is not a mind product, but rather a pre-existing and complete entity which we humans (and dogs) only discover. If this were true, we could expect this “perfect” mathematics to be… well… perfect, i.e., that it’s correspondence to nature (i.e., material reality) would be absolute, but this is not the case as is illustrated by it’s inability to give us a firm answer on the circumference of a circle when all we know is the diameter. Also, let’s not forget Godel’s incompleteness theorem, not to mention Turing and Chaitin.
Math is an excellent tool. We humans get everything we need from it. (Pi can give us the circumference of a circle to any degree of precision we need; it just can’t give us an absolute answer which, for the most part, we don’t need.) The math which emerges from our brains is pragmatic because as an evolutionary product it contributes to our survival, but not necessarily precisely to our understanding of the universe because we more or less don’t need that.
I guess this is the stuff for a book, but I’m just an amateur in these things, and these are my amateurish thoughts on the subject. All that I’ve said so far on these things is simply my own view, developed during just a couple of years of retirement from a life of work in a rather different area. I’d like to keep discussing this with those of you who know more about this, or maybe know less, but who like to think about this kind of thing.
Ray
Some would assert that mathematics has an independent existence, that it is not a mind product, but rather a pre-existing and complete entity which we humans (and dogs) only discover. If this were true, we could expect this “perfect” mathematics to be… well… perfect, i.e., that it’s correspondence to nature (i.e., material reality) would be absolute, but this is not the case as is illustrated by it’s inability to give us a firm answer on the circumference of a circle when all we know is the diameter. Also, let’s not forget Godel’s incompleteness theorem, not to mention Turing and Chaitin.
Math is an excellent tool. We humans get everything we need from it. (Pi can give us the circumference of a circle to any degree of precision we need; it just can’t give us an absolute answer which, for the most part, we don’t need.) The math which emerges from our brains is pragmatic because as an evolutionary product it contributes to our survival, but not necessarily precisely to our understanding of the universe because we more or less don’t need that.
I guess this is the stuff for a book, but I’m just an amateur in these things, and these are my amateurish thoughts on the subject. All that I’ve said so far on these things is simply my own view, developed during just a couple of years of retirement from a life of work in a rather different area. I’d like to keep discussing this with those of you who know more about this, or maybe know less, but who like to think about this kind of thing.
Ray
2bertilak
Recommended reading: Where mathematics comes from by George Lakoff.
3jeffoz
The beginnings of mathematics is discussed in Umberto Eco's On Beauty: A History of a Western Idea.
As the subhead implies, we are in the world if ideas,initially Pythogorean later to become Plato's "Forms"; and still later Descartes philosophical musings. All of which have had an enormous influence on Western philosophy.
An interesting alternative view is that of Merleau Ponty's Phenomenology; especially The Visible and the Invisible, a good introduction might be Ponty's The Phenomenology of Perception or an excellent overview by Lawrence Hass, Merleau Ponty's Philosophy.
My take is that the dog's ability is a capacity that is a developed from an inherent potential that increases with practice, just as maths is developed by doing. Your description of the dog's ability in terms of maths is, in my opinion, a second order category. Perception and being in the world comes first, maths and science while valuable in their own right are man-made creations. To seek an explanation from secondary sources, will take you through all the history of Western philosophy since Thales - an immense, though fascinating task (which I've tried with limited success). To be able to see the big picture from two points of view may give a more balanced view. In the end none are in the domain of certainty; it is our curse or blessing to be questioning beings. No one who lives in this world is an amatuer, though some may be more endowed intellectually than others. Like you, I like thinking about these things as well, whether well or ill may be left to others to judge.
Geoff
As the subhead implies, we are in the world if ideas,initially Pythogorean later to become Plato's "Forms"; and still later Descartes philosophical musings. All of which have had an enormous influence on Western philosophy.
An interesting alternative view is that of Merleau Ponty's Phenomenology; especially The Visible and the Invisible, a good introduction might be Ponty's The Phenomenology of Perception or an excellent overview by Lawrence Hass, Merleau Ponty's Philosophy.
My take is that the dog's ability is a capacity that is a developed from an inherent potential that increases with practice, just as maths is developed by doing. Your description of the dog's ability in terms of maths is, in my opinion, a second order category. Perception and being in the world comes first, maths and science while valuable in their own right are man-made creations. To seek an explanation from secondary sources, will take you through all the history of Western philosophy since Thales - an immense, though fascinating task (which I've tried with limited success). To be able to see the big picture from two points of view may give a more balanced view. In the end none are in the domain of certainty; it is our curse or blessing to be questioning beings. No one who lives in this world is an amatuer, though some may be more endowed intellectually than others. Like you, I like thinking about these things as well, whether well or ill may be left to others to judge.
Geoff
