G\u00f6del’s Incompleteness Theorem shows that human beings can never formulate a correct and complete description of the set of natural numbers, {0, 1, 2, 3, . . .}. But if mathematicians cannot ever fully understand something as simple as number theory, then it is certainly too much to expect that science will ever expose any ultimate secret of the universe.<\/p>\n
Scientists are thus left in a position somewhat like K. in The Castle [of Kafka]. Endlessly we hurry up and down corridors, meeting people, knocking on doors, conducting our investigations. But the ultimate success will never be ours. Nowhere in the castle of science is there a final exit to absolute truth…<\/p>\n
There is one idea truly central to G\u00f6del\u2019s thought that we discussed at some length. This is the philosophy underlying G\u00f6del\u2019s credo, \u201cI do objective mathematics.\u201d By this, G\u00f6del meant that mathematical entities exist independently of the activities of mathematicians, in much the same way that the stars would be there even if there were no astronomers to look at them. For G\u00f6del, mathematics, even the mathematics of the infinite, was an essentially empirical science.<\/p>\n
According to this standpoint, which mathematicians call Platonism, we do not create the mental objects we talk about. Instead, we find them, on some higher plane that the mind sees into, by a process not unlike sense perception.<\/p>\n
The philosophy of mathematics antithetical to Platonism is formalism, allied to positivism. According to formalism, mathematics is really just an elaborate set of rules for manipulating symbols. By applying the rules to certain \u201caxiomatic\u201d strings of symbols, mathematicians go about \u201cproving\u201d certain other strings of symbols to be \u201ctheorems.\u201d<\/p>\n
The game of mathematics is, for some obscure reason, a useful game. Some strings of symbols seem to reflect certain patterns of the physical world. Not only is \u201c2 + 2 = 4\u201d a theorem, but two apples taken with two more apples make four apples.<\/p>\n
It is when one begins talking about infinite numbers that the trouble really begins. Cantor\u2019s Continuum Problem is undecidable on the basis of our present-day theories of mathematics. For the formalists this means that the continuum question has no definite answer. But for a Platonist like G\u00f6del, this means only that we have not yet \u201clooked\u201d at the continuum hard enough to see what the answer is.<\/p>\n
In one of our conversations I pressed G\u00f6del to explain what he meant by the \u201cother relation to reality\u201d by which he said one could directly see mathematical objects. He made the point that the same possibilities of thought are open to everyone, so that we can take the world of possible forms as objective and absolute. Possibility is observer-independent, and therefore real, because it is not subject to our will.<\/p>\n
There is a hidden analogy here. Everyone believes that the Empire State Building is real, because it is possible for almost anyone to go and see it for himself. By the same token, anyone who takes the trouble to learn some mathematics can \u201csee\u201d the set of natural numbers for himself. So, G\u00f6del reasoned, it must be that the set of natural numbers has an independent existence, an existence as a certain abstract possibility of thought.<\/p>\n
I asked him how best to perceive pure abstract possibility. He said three things, i) First one must close off the other senses, for instance, by lying down in a quiet place. It is not enough, however, to perform this negative action, one must actively seek with the mind, ii) It is a mistake to let everyday reality condition possibility, and only to imagine the combinings and permutations of physical objects\u2014the mind is capable of directly perceiving infinite sets, iii) The ultimate goal of such thought, and of all philosophy, is the perception of the Absolute.<\/p>\n
G\u00f6del rounded off these comments with a remark on Plato: \u201cWhen Plautus could fully perceive the Good, his philosophy ended.\u201d<\/p>\n
G\u00f6del shared with Einstein a certain mystical turn of thought. The word \u201cmystic\u201d is almost pejorative these days. But mysticism does not really have anything to do with incense or encounter groups or demoniac possession. There is a difference between mysticism and occultism.<\/p>\n
A pure strand of classical mysticism runs from Plato to Plotinus and Eckhart to such great modern thinkers as Aldous Huxley and D. T. Suzuki. The central teaching of mysticism is this: Reality is One. The practice of mysticism consists in finding ways to experience this higher unity directly.<\/p>\n
The One has variously been called the Good, God, the Cosmos, the Mind, the Void, or (perhaps most neutrally) the Absolute. No door in the labyrinthine castle of science opens directly onto the Absolute. But if one understands the maze well enough, it is possible to jump out of the system and experience the Absolute for oneself.<\/p>\n
The last time I spoke with Kurt G\u00f6del was on the telephone, in March 1977. I had been studying the problem of whether machines can think, and I had become interested in the distinction between a system\u2019s behavior and the underlying mind or consciousness, if any.<\/p>\n
What had struck me was that if a machine could mimic all of our behavior, both internal and external, then it would seem that there is nothing left to be added. Body and brain fall under the heading of hardware. Habits, knowledge, self-image and the like can all be classed as software. All that is necessary for the resulting system to be alive is that it actually exist.<\/p>\n
In short, I had begun to think that consciousness is really nothing more than simple existence. By way of leading up to this, I asked G\u00f6del if he believed there is a single Mind behind all the various appearances and activities of the world.<\/p>\n
He replied that, yes, the Mind is the thing that is structured, but that the Mind exists independently of its individual properties.<\/p>\n
I then asked if he believed that the Mind is everywhere, as opposed to being localized in the brains of people.<\/p>\n
G\u00f6del replied, \u201cOf course. This is the basic mystic teaching.\u201d<\/p>\n
We talked a little set theory, and then I asked him my last question: \u201cWhat causes the illusion of the passage of time?\u201d<\/p>\n
G\u00f6del spoke not directly to this question, but to the question of what my question meant\u2014that is, why anyone would even believe that there is a perceived passage of time at all.<\/p>\n
He went on to relate the getting rid of belief in the passage of time to the struggle to experience the One Mind of mysticism. Finally he said this: \u201cThe illusion of the passage of time arises from the confusing of the given with the real. Passage of time arises because we think of occupying different realities. In fact, we occupy only different givens. There is only one reality.\u201d<\/p>\n
__________<\/p>\n
Excerpts from Rudy Rucker’s, Memories of Kurt G\u00f6del, Rudy’s Blog<\/a>, first appeared in the magazine Science 82 in April 1982, and in Rudy’s 1982 book Infinity and the Mind<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":" G\u00f6del’s Incompleteness Theorem shows that human beings can never formulate a correct and complete description of the set of natural numbers, {0, 1, 2, 3, . . .}. But if mathematicians cannot ever fully understand something as simple as number theory, then it is certainly too much to expect that science will ever expose any […]<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_disable_autopaging":false},"categories":[4],"tags":[1607,5733,5735,5734,5732,210,5683,30,110],"class_list":["post-2470","post","type-post","status-publish","format-standard","hentry","category-plato","tag-eckhart","tag-epistemology","tag-goedel","tag-mathematics","tag-model-theory","tag-mysticism","tag-plato","tag-platonism","tag-plotinus"],"_links":{"self":[{"href":"https:\/\/www.ellopos.com\/blog\/wp-json\/wp\/v2\/posts\/2470","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ellopos.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ellopos.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ellopos.com\/blog\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ellopos.com\/blog\/wp-json\/wp\/v2\/comments?post=2470"}],"version-history":[{"count":0,"href":"https:\/\/www.ellopos.com\/blog\/wp-json\/wp\/v2\/posts\/2470\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.ellopos.com\/blog\/wp-json\/wp\/v2\/media?parent=2470"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ellopos.com\/blog\/wp-json\/wp\/v2\/categories?post=2470"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ellopos.com\/blog\/wp-json\/wp\/v2\/tags?post=2470"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}