Forms, Logic, Mathematics and the "F" Meme
"Laws of Form" are the foundation for mathematics based on the fundamental idea of categorization. Any freshman course will show how all of mathematics is built up on the foundation of set theory - the logic of categories. Even in High School, we are taught that there is no such thing as a "real" circle - in the real world, things that we call circles are actually only approximately circular. But we are left with the idea that a circle is a "real thing". "More real", in fact, than the circles of actual experience.
Most people get scared off of mathematics and logic before they start to ask questions about the relationship between "math" and the real world. This is aided and abetted by the tendency of editors to banish all mathematical formulas from the pages of popular books and even magazines devoted to Science (Such as Scientific American).
For those on a spiritual path, Zen (and its de-mystified version, Mindfulness) points out how categories are arbitrary creations of the mind. The horrendous implications of this (that Math and Logic are also arbitrary) are usually glossed over.
But for those such as myself, who have spent years exploring the dark corridors of mathematics and "pure" logic, the inevitable question emerges: Is this world "real" or is it "just a construction of the human mind"? Such questions usually occur to those who are also on the "spiritual path". That path leads through the observation that everything is a creation of the mind, but there certainly seems to be more "out there" than what is running around between my ears. Logic seems to be a very solid category of reality in addition to mind and the real world. Even aliens from another dimension would understand basic logic and math. Wouldn't they?
"Applied Mathematics", including some of its rather obscure branches, such as group theory and topology, have found to be amazingly useful in describing phenomena of the "real world". You get the impression that mathematical principles must underlie reality itself. In fact, it's quite easy to imagine alternate universes with quite different "laws" that are still bound to obey the rules of mathematics. Einstein, said that the most amazing thing about the universe is how much of its workings can be understood by the human mind. But he didn't say it could all be understood.
Undermining our confidence is the EPR paradox, which seems to indicate that our fundamental ideas of logic break down when confronted with some relatively simple properties of the universe. This is actually the point where Einstein's famous "thought experiments" mislead him - he was wrong in his confidence that properties of sub-atomic particles "must" be "out there in the real world", independent of our measurements. Our response, of course, is always to invent new math and logic to describe the paradoxical situation, with confidence that logic must, in the end, be preserved. The universe itself "must" be logical.
The idea that mathematics is somehow "more real" than reality itself goes all the way back to Plato, who considered "forms" to be more real than any particular examples of forms. If we could only see with the proper eyes, we'd see only the celestial reality of ultimate forms. This amounts to saying that the "form" is ultimate reality. A "form" is a category. Hofstadter has finally buried Plato's insight by pointing out how arbitrary and "fuzzy" our categories are - hinting that categories are an inescapable artifact of the human mind.
Accepting Hofstadter's perspective, while also respecting the stunning applicability of the concept of forms, logic and mathematics, we are lead to the ultra-powerful "F" meme, which includes trust in our categories, our logic and our math as a valid description of the real world, while also recognizing that "F" is just another "meme". It should take its place along the other "big ones": mind and reality.
Trust in the "F" meme can assume religious overtones. The reductionist view can take mathematics and logic as "fundamental" reality. Serious modern intellectuals can claim that there is no real world except what our observations (part of the "M" world) plus our logic (The "F" world) tell us. The religious nature of this outlook is discussed here. The "religion" described owes more to Plato (mentioned above) than the Bible, but it's also true that the New Testament was strongly influenced by Greek philosophy, which would readily accept the idea of a heavenly realm, more "real" than the one we inhabit. In the Old Testament, God seems to live above the sky, not in some alternate reality.
We take the form of distinction for the form." There is a circularity in bringing into words what is quite clear without them. And yet it is in the bringing forth into formalisms that mathematics is articulated and universes of discourse come into being. The elusive beginning, before there was a difference, is the eye of the storm, the calm center from which these musings spring.There is a delicious circularity in this idea - something that fans of Hofstadter will recognize as the fingerprint of the human mind at its best.
Most people get scared off of mathematics and logic before they start to ask questions about the relationship between "math" and the real world. This is aided and abetted by the tendency of editors to banish all mathematical formulas from the pages of popular books and even magazines devoted to Science (Such as Scientific American).
For those on a spiritual path, Zen (and its de-mystified version, Mindfulness) points out how categories are arbitrary creations of the mind. The horrendous implications of this (that Math and Logic are also arbitrary) are usually glossed over.
But for those such as myself, who have spent years exploring the dark corridors of mathematics and "pure" logic, the inevitable question emerges: Is this world "real" or is it "just a construction of the human mind"? Such questions usually occur to those who are also on the "spiritual path". That path leads through the observation that everything is a creation of the mind, but there certainly seems to be more "out there" than what is running around between my ears. Logic seems to be a very solid category of reality in addition to mind and the real world. Even aliens from another dimension would understand basic logic and math. Wouldn't they?
"Applied Mathematics", including some of its rather obscure branches, such as group theory and topology, have found to be amazingly useful in describing phenomena of the "real world". You get the impression that mathematical principles must underlie reality itself. In fact, it's quite easy to imagine alternate universes with quite different "laws" that are still bound to obey the rules of mathematics. Einstein, said that the most amazing thing about the universe is how much of its workings can be understood by the human mind. But he didn't say it could all be understood.
Undermining our confidence is the EPR paradox, which seems to indicate that our fundamental ideas of logic break down when confronted with some relatively simple properties of the universe. This is actually the point where Einstein's famous "thought experiments" mislead him - he was wrong in his confidence that properties of sub-atomic particles "must" be "out there in the real world", independent of our measurements. Our response, of course, is always to invent new math and logic to describe the paradoxical situation, with confidence that logic must, in the end, be preserved. The universe itself "must" be logical.
The idea that mathematics is somehow "more real" than reality itself goes all the way back to Plato, who considered "forms" to be more real than any particular examples of forms. If we could only see with the proper eyes, we'd see only the celestial reality of ultimate forms. This amounts to saying that the "form" is ultimate reality. A "form" is a category. Hofstadter has finally buried Plato's insight by pointing out how arbitrary and "fuzzy" our categories are - hinting that categories are an inescapable artifact of the human mind.
Accepting Hofstadter's perspective, while also respecting the stunning applicability of the concept of forms, logic and mathematics, we are lead to the ultra-powerful "F" meme, which includes trust in our categories, our logic and our math as a valid description of the real world, while also recognizing that "F" is just another "meme". It should take its place along the other "big ones": mind and reality.
Trust in the "F" meme can assume religious overtones. The reductionist view can take mathematics and logic as "fundamental" reality. Serious modern intellectuals can claim that there is no real world except what our observations (part of the "M" world) plus our logic (The "F" world) tell us. The religious nature of this outlook is discussed here. The "religion" described owes more to Plato (mentioned above) than the Bible, but it's also true that the New Testament was strongly influenced by Greek philosophy, which would readily accept the idea of a heavenly realm, more "real" than the one we inhabit. In the Old Testament, God seems to live above the sky, not in some alternate reality.
Comments
Post a Comment