Gödel And The Limits of Rationality
From the conclusion of “Cantor’s Diagonalization Method” by Alexander Kharazishvili (and I’ll assume the reader has some familiarity with Gödel’s … Continue Reading Gödel And The Limits of Rationality
Gödel And The Limits of Rationality
From the conclusion of “Cantor’s Diagonalization Method” by Alexander Kharazishvili (and I’ll assume the reader has some familiarity with Gödel’s famous theorems):
Tarski’s theorem leads to Gödel’s first incompleteness theorem. Let Pr denote the set of natural numbers corresponding to the provable sentences in formal arithmetic. Pr is recursively enumerable. Since any recursively enumerable set of integers is arithmetical, Pr cannot coincide with Tr [“the set of all natural numbers that correspond to those sentences true in the standard model of formal arithmetic”]. It can only be a proper subset of Tr. Truth and provability are not the same. That this discovery was revolutionary hardly does justice to its significance. Thus, there is a sentence S that is true, but not provable, in the standard model of formal arithmetic. Neither S, nor its negation, ¬S, is provable and formal arithmetic is incomplete. A similar argument is applicable to any recursive mathematical theory that is stronger than formal arithmetic (e.g., ZFC set theory).
We’ve talked about this many times: the idea that reason and rationality are enough, or are the ultimate (as in best) form of thought, are obviously false ideas. They are useful and necessary, but alone they are insufficient.
Reason, in the way I am thinking about it, is almost mechanistic. It takes a pile of thoughts, processes them by known rules, and spits out ideas. The processing is like a machine, which can be souped up or in poor repair. However complex it is, it isn’t anything more than brute mental force.
The ore at the base of the pile of thoughts must be supplied from outside. Reason needs fuel it can’t provide itself. Reason operating only itself is like a perpetual motion machine, an impossibility.
It is not reason that concludes Reason is reasonable. It is not Reason that proves truths that cannot be proved by Reason. “Truth and provability are not the same.” Provability by Reason, that is. Proof by other mechanisms is still possible; indeed, necessary.
All mathematics proceeds from two things: (1) unproveable-by-Reason propositions (axioms) and (2) unproveable-by-Reason belief Reason will work to generate new true propositions. This sequence does not apply to mathematics alone, but all thought.
Therefore there must be something beyond Reason providing our deepest and most consequential truths.
Amazon’s AI Proves Men Better Than Women At Tech Jobs
Item Amazon scraps secret AI recruiting tool that showed bias against women (Thanks to Mark Charters for the tip.)
Amazon.com Inc’s (AMZN.O) machine-learning specialists uncovered a big problem: their new recruiting engine did not like women.
The team had been building computer programs since 2014 to review job applicants’ resumes with the aim of mechanizing the search for top talent, five people familiar with the effort told Reuters…
“Everyone wanted this holy grail,” one of the people said. “They literally wanted it to be an engine where I’m going to give you 100 resumes, it will spit out the top five, and we’ll hire those.”
But by 2015, the company realized its new system was not rating candidates for software developer jobs and other technical posts in a gender-neutral way.
That is because Amazon’s computer models were trained to vet applicants by observing patterns in resumes submitted to the company over a 10-year period. Most came from men, a reflection of male dominance across the tech industry.
We’ve seen before that “algorithms” are called “racist.” Read it.
Feed the algorithm, the curve-fitting AI, measures such as use of purple hair dye, purchased tampons or bought video game, and so forth, and it will, for the painfully obvious reasons, pick out men from women. Not with perfection, of course, but it will be pretty good.
Then, since as everybody knows, but many don’t like knowing, men at the extremes are better at analytic tasks than non-men, an algorithm to maximize a candidate’s ability to code not fed sex, but measures highly predictive of sex, will pick out more men than women. The algorithm will be “biased” (to reality).
There are only two ways to avoid the algorithm suggesting more men than women: (1) feed the algorithm only measures which in no way are predictive of sex; but, since men (at the extremes) are better than non-men at coding, the algorithm will do a lousy job predicting coding success; (2) instruct the algorithm to spit out Equality; which also will force the algorithm to do a rotten job.
Equality is defined as the hope in absence of all evidence that men and women are equal. But if men and women were equal, we would not even know to say “men” and “women.”
Bias is defined as politically unacceptable result.
Summary Against Modern Thought: You Need God To See God
You can’t do everything on your own: be sure to first review last week for an understanding of what that means.
THAT NO CREATED SUBSTANCE CAN, BY ITS OWN NATURAL POWER, ATTAIN THE VISION OF GOD IN HIS ESSENCE
1 However, it is not possible for any created substance, by its own power, to be able to attain this manner of divine vision.
2 Indeed, a lower nature cannot acquire that which is proper to a higher nature except through the action of the higher nature to which the property belongs. For instance, water cannot be hot except through the action of fire. Now, to see God through His divine essence is proper to the divine nature, for it is the special prerogative of any agent to perform its operation through its own form. So, no intellectual substance can see God through His divine essence unless God is the agent of this operation.
Notes Hence, “if you seek Him you have already found him.”
3 Again, the form proper to any being does not come to be in another being unless the first being is the agent of this event, for an agent makes something like itself by communicating its form to another thing. Now, it is impossible to see the substance of God unless the divine essence itself is the form whereby the intellect understands, as we have proved. Therefore, it is not possible for a created substance to attain this vision, except through divine action.
4 Besides, if any two factors are to be mutually united, so that one of them is formal and the other material, their union must be completed through action coming from the side of the formal factor, and not through the action of the one that is material. In fact, form is the principle of action, while matter is the principle of passion. For the created intellect to see God’s substance, then, the divine essence itself must be joined as an intelligible form to the intellect, as we have proved. Therefore, it is not possible for the attainment of this vision to be accomplished by a created intellect except through divine action.
Notes This ties to what we have been saying about the highest forms of knowledge, i.e. that which comes directly from God, as insight or (sometimes) intuition. I.e. induction, of a sort.
5 Furthermore, “that which is of itself is the cause of that which is through another being.” But the divine intellect sees the divine substance through itself, for the divine intellect is the divine essence itself whereby the substance of God is seen, as was proved in Book One [45]. However, the created intellect sees the divine substance through the essence of God, as through something other than itself. Therefore, this vision cannot come to the created intellect except through God’s action.
6 Moreover, whatever exceeds the limitations of a nature cannot accrue to it except through the action of another being. For instance, water does not tend upward unless it is moved by something else. Now, seeing God’s substance transcends the limitations of every created nature; indeed, it is proper for each created intellectual nature to understand according to the manner of its own substance. But divine substance cannot be understood in this way, as we showed above. Therefore, the attainment by a created intellect to the vision of divine substance is not possible except through the action of God, Who transcends all creatures.
7 Thus, it is said: “The grace of God is life everlasting” (Rom. 6:23). In fact, we have shown that man’s happiness, which is called life everlasting, consists in this divine vision, and we are said to attain it by God’s grace alone, because such a vision exceeds all the capacity of a creature and it is not possible to reach it without divine assistance. Now, when such things happen to a creature, they are attributed to God’s grace. And the Lord says: “I will manifest Myself to him” (John 14:21).
Insanity & Doom Update LX
Item Item removed! I was too eager for Doom and was hoaxed. Pure laziness on my part. Well, partial stupidity, too. Thanks to Valeriu for the tip. See the comment below. As a non-Doom compensation, I present this commercial.
Item How a Social Justice Mob Fired a Tenured Professor
The fall semester is off to a fiery start. We have Brown University’s decision to distance itself from Professor Lisa Littman’s research paper; the decision by the New York Journal of Mathematics journal to un-publish Professor Theodore Hill’s study; the University of Chicago’s refusal to defend Professor Rachel Fulton Brown from scurrilous attack led by a Brandeis professor; and the rush to give NYU Professor Avital Ronell a free pass for having harassed and sexually assaulted a gay graduate student…
Political correctness continues to explore new frontiers. The common theme in these four cases is the supreme confidence of the academic left. It quashes any views it dislikes without a moment’s hesitation and feels little call to explain or justify its actions. Rather, it relies on the readiness of the academy at large to applaud the effort to keep thinkers-of-dangerous-thoughts in their cages….
The Canadian case of the moment involves a tenured associate professor of psychology at Acadia University in Nova Scotia. Professor Rick Mehta was suddenly fired from his position on August 31. The stated reason, provided to Professor Mehta in a letter from President Peter Ricketts, was: “failure to fulfill [his] academic responsibilities, unprofessional conduct, breach of privacy, and harassment and intimidation of students and other members of the University community.”…
No, rather, [Meta] described multiculturalism as a “scam.” Multiculturalism might be described as the official state religion of Canada, and Canadian universities as its schools of theology. The courage to call it out as a scam testifies that Professor Mehta must be a man of rare character. Let me say at once that I have never met him or even corresponded with him, and it is possible that he holds other opinions from which I would recoil in horror. But his stand on multiculturalism all by itself commands respect.
This story is from Peter Wood, the National Association of Scholar’s head, and should be read in full. Any stand against multiculturalism does indeed command respect. What’s odd about this story, though, is that Meta, as his name indicates, is of Indian descent.
Anyway, here is the tale of Rachel Fulton Brown spoken of above, another tenured professor under fire for speaking of Reality. Also from Wood.
The NAS, God bless them all, think universities may be saved. I pray they are right. I believe they are wrong. But that doesn’t mean the NAS doesn’t deserve our support. We have to make the Left pay for every inch of ground they take—until such time as we can fall back.
And nuke ’em from orbit. It’s the only way to be sure.