#on logic

[thequotablescollector]

"If one person believes something illogical, he is called a fool – but if ten million people believe the same illogical thing, it is called religion." – Voltaire

[floatingbook]

On men, male-partnered ‘feminists’, and logic

- A lesbian perspective. (09/11/2020)

Just so we are clear about terms here, women are adult human females. Men are adult human males. Feminism is a movement fuelled by theory aimed at women’s liberation. Liberation means that we have to be liberated from something. If you have eyes and ears and are reading this, you are probably aware that women are consistently discriminated against on the basis of their sex by men and the institutions they have built and perpetuate, by among others (not an exhaustive list) of sexual assault and rape, control over our bodies and their reproductive capacities, theological lowering, persecution of lesbians, diminished economic and financial prospects, prostitution, pornography, the enforcing of strict norms over our existences, or plain murder. Men are doing this to us. Men are the problem we face if as women we want to live our lives free from oppression.

So why would women who are well aware of the source of our continued misery (men, if you needed a reminder) still want to date men? In the world we live in, there is no man devoid of misogyny (and if there was one, he would not be dating you, he would be spending all his waking hours trying to reform his fellow men, as anything less given the state of things is misogyny). So if a woman wants to date a man, knowing what she knows about men being constant oppressors of women, she has to not care about his misogyny. She knows, and yet she says “I don’t mind”, “It’s not enough of a deal-breaker”. How do you set the bar as to the level of misogyny you will tolerate? Is it enough if he doesn’t beat and rape you? Don’t you mind having to clean up after him all the time, being expected to do all the dishes, gaining no financial independence beyond some support as long as he tolerates you, his consumption of porn, his potential molestation of your hypothetical children? Is it fine if he is only a misogynist to other women and not you, talking over his female colleagues, doing nothing about the skewered wage gap in his favour, never voting for a woman in elections, saying nothing when other men are being predators? How do you make peace with the fact that you are reinforcing men’s expectation of a sexual partner and, quite often, maid as something they deserve just by being alive and breathing men? How do you enter into a partnership with a man knowing that in doing so, you’re upholding the patriarchal mean of control that is marriage, meant to pass women from father to husband as pieces of property, an institution that will do its best to sever your links to other women? No man is special; no man is free of misogyny.

How do you call yourself a ‘radical feminist’ and say that male-attracted women shouldn’t be expected not to date, partner with or marry men, if we ever are to achieve women’s liberation? What is your logic? Are you saying that women cannot be expected to survive without regular sex? In which case, how can you then tell ‘incels’ that they are not entitled to a woman’s body? Because your logic then is that sex is a need. Or are you saying that women can have no life without a man`? In that case, why are you even concerned by women’s liberation? By your logic, it would kill all male-attracted women. Or are you saying that women can have no fulfilling life without a man? In that case, maybe talk to all the celibate male-attracted women who are celibate because they have realised that men are nothing but a danger to us, or to any celibate woman really. Our lives are good. Plus, do you see how reductive that reasoning is? You can have a perfectly fine and happy and fulfilling life without tying yourself up to a man. You are a full person in your own right. You don’t need a man in your life to have succeeded at at the game called existence. Especially given the circumstances and the fact that any men will likely make your life worse.

So if you know, why do you insist that choosing to date men is anything other that a negative choice for yourself and the women around you? Sure, we don’t make our choices in a vacuum, and for most women that means never questioning the logic behind dating men. But we also don’t make our choices in a vacuum, and that means that if you do know what your decision entails, you are supporting misogyny and patriarchy by dating men. Heterosexual partnerships are not neutral relationships in the world we live in. Because in most countries, and certainly for most of the women reading this, women can now have their own finances and are no longer forced into marriages to survive. What is your excuse? Because, while you would not be to blame if a man abused you, if you are aware of the danger, you should act to prevent that danger of harming you. Think of it like vaccines; the probability that you will ever encounter some of the diseases we get vaccinated against are extremely low, but we still get vaccinated to prevent harm. Maybe he won’t abuse you, maybe he won’t be an overt misogynist to you, but he will still be a misogynist. So why risk it? Why? What’s your logic, when you’re on the other hand pretending to want women’s liberation?

Note, before anyone starts clowning, that lesbians criticising heterosexual partnerships in relation to women’s oppression are not advocating for male-attracted women to date us instead or being jealous because they are single. We are tired of the constant misogyny and homophobia in our lives and are trying to do something about it.

[rxshl]

ON LOGIC + MATH AS AN INVENTION VS DISCOVERY + GOLDBACH x GÖDEL

In formal systems of logical reasoning there are two kinds of statements: axioms, known or assumed to be true, and theorems, which require proof to be established. For instance, in arithmetic, a primary, simple axiom is the “reflexive axiom,” which states that a = a for any numerical value substituted for the symbol or variable a. So, 3 = 3. And 156,033,041 = 156,033,041.

Another fundamental axiom is the “symmetric axiom,” which states that things on opposite sides of an equal sign are the same. So if a = b, then b = a. The “transitive axiom” states that if a = b and b = c, then a = c, an equivalent to Euclid’s statement of geometry: “Things that are equal to the same thing are also equal to one another.” These statements are considered true on their face and require no proof.

Theorems, however, may or may not be true. Theorems, like hypotheses in scientific theories, require proof. A well-stated theorem may appear to be true, but one can’t assume that it is. One has to prove the truth of a new theorem by starting with the foundational axioms of the system and using them to ascend methodically, line by line, rung by rung, up the logical ladder of the proof. When the new theorem has been arrived at, it is considered proved. Alternatively, if the proof leads to the opposite of the new theorem, then the theorem has been disproved.

Sometimes theorems seem obvious, but proving them is enormously difficult. For example, there is the famous Goldbach conjecture, which states that every even whole number greater than 2 is the sum of at least one pair of prime numbers. Take the number 8; it is the sum of 3 + 5, both primes. There are a whole bunch of pairs of primes that add up to 144, including 97 + 47, 103 + 41, and 139 + 5. The Goldbach conjecture has been shown to be true by laborious calculations by hand for numbers up to 100,000 and then by computers up to 4 × 1017. But these are not proofs of the conjecture, always true no matter how high one goes; they are simply a lot of calculations. We remain uncertain whether we might eventually find an even larger number that would be an exception. The Goldbach conjecture remains unproved to this day.

In 1920, the great German mathematician David Hilbert announced a program that itemized what he considered to be the most important challenges for setting mathematics on firm foundations. Hilbert supported the use of a “formal language” of symbols to write mathematical statements in proofs. The success and validity of any axiomatic system and the proofs of its many theorems, he said, were to be judged on particular criteria. These criteria stated that a system must be consistent within itself, meaning that it cannot paradoxically contradict itself by simultaneously proving some theorem to be both true and not true. The system also has to be complete, meaning that it has to have within itself the means of proving that every true statement about the system is indeed true. The system that makes up arithmetic, for example, must contain within itself the means to prove every true statement about arithmetic, even the Goldbach conjecture. Consistency and completeness were now the seals of success for any mathematical system. The Vienna Circle’s mission was completely in accord with Hilbert’s program.

Enter the small, fine-featured, bespectacled figure of a still young Kurt Gödel, who would one day become known as the greatest logician since Aristotle, if not, indeed, the greatest of all time. He sat quietly in the back of the Vienna Circle meetings, keeping his own counsel, his life’s pattern already set: to withhold comment or commentary until his answer was perfect, polished, and definitive. We can imagine his head turning this way and that, closely following the back-and-forth of his colleagues’ debates, like the pendulum of a clock.

It was the Vienna Circle’s clock that was ticking.

Gödel arrived in Vienna from Brünn, the city of his birth, in 1924, having already mastered university-level mathematics at eighteen years old. It was in these early years at the University of Vienna that he encountered the ideas that would lead him to embrace mathematical Platonism. Mathematical Platonists believe that mathematical expressions—numbers and formulas and geometric forms—belong to the realm of Plato’s ideals, not to the realm of material existence.

From this view, mathematics is not merely a means invented by humans to count bushels of wheat; it is a true realm unto itself, beyond our own questing human minds. Mathematics awaits human discovery, not human invention. Euclidean geometry (with equations like the Pythagorean theorem), Newton’s calculus for describing fluid motion, the wave equations of Schrödinger, the Mandelbrot set: these were not inventions; they were discoveries.

In contrast, for the Vienna Circle, numbers and forms of mathematics were the logical creations of human minds—invented, not discovered—purely tools for describing physical reality. For them, mathematics derived logically through human innovation, from the simple “real” numbers of counting and the simple geometries of Euclid.

When Gödel was ready to detonate the Vienna Circle’s plans with his own “incompleteness proofs,” he did so quietly, in an almost offhand manner.

The proofs of his two “incompleteness theorems” are widely recognized as a dazzling display of intuitive genius, their mathematical beauty often compared to that of Bach’s most complex musical canons or the elaborate architecture of Gothic cathedrals. The detailed methods of the first proof are well beyond the scope of this book, but their ingenuity can be conveyed.

Gödel’s intuition was that there would be statements about formal systems of arithmetic that were true, but that could not be proved to be true from within arithmetic’s own axioms and theorems. This was a Platonic perspective: mathematical truths simply exist “out there” in the realm of the ideal, awaiting our discovery. There is nothing, however, to require that every single mathematical statement must be amenable to our proofs. It was only the hubris of formalist mathematicians that said otherwise.

So, if Gödel could prove that there were statements that were in fact true, but could not be proved as such, he would then prove that Hilbert was wrong: incompleteness could not be completely eliminated from mathematics. The question was how to do so.

Gödel cleverly devised a numbering system whereby each of the thirteen symbols used in constructing a logical statement in a proof could be substituted by a number (1 through 13) and the logical statement as a whole could thereby, through a procedure Gödel designed, be converted into a unique number not shared by any other formal statement. The code was bidirectional: not only did every logical statement have a unique number, but any number could in turn be decoded to reveal the unique formal set of symbols of an underlying logical statement.

Through this ingenious numbering system, sequential statements within the proof had both a purely arithmetical relationship, as well as a logical one. Gödel’s proof was therefore metamathematical: a proof composed of the very things with which the proof was concerned, numbers. The arithmetical relationship between the representative numbers conveyed arithmetical truths that were parallel to the step-by-step logic of the proof. So logical statements could be about numbers, but numbers, in turn, could convey logical statements. Following Gödel’s self-referential, looping logic is like treading a path along a Möbius strip, round and round.

As if this cleverness weren’t enough, in the next step Gödel’s ingenuity truly soared. He created a logical statement (again using the symbols of formal logic that could be replaced by Gödel numbering) that, in English, asserts something like: “This statement cannot be proved from within this system.”

This is a classical paradoxical statement, much like the liar’s paradox, attributed to Epimenides the Cretan, which has been pondered for centuries. “All Cretans are liars” is problematic since, if it is true, then the Cretan saying it is lying, in which case it is false. If it is false, then it is a lie, in which case the statement is true: Cretans are always liars. Round and round, like a snake swallowing its tail.

Here is another version, called the “card paradox,” by the logician Philip Jourdain, a student of the renowned mathematician Bertrand Russell: if one writes on a slip of paper, “The statement on the other side of this paper is false,” and, on the other side, one writes, “The statement on the other side of this paper is true,” then we get into a similar endless roundabout.

Gödel did not fear paradox; he welcomed it. His special statement—“This statement cannot be proved from within this system”—takes precisely the same endlessly circular form. If the statement can be proved from within the logical system, then the statement is false. If it is false, then it can’t be proved, in which case it is true and is proved.

The step he took next is both simple and breathtaking. A clear, informal description of it is provided by science writer James Gleick: “Gödel showed how to construct a formula that said A certain number, x, is not provable. That was easy: there were infinitely many such formulas. He then demonstrated that, in at least some cases, the number x would happen to represent that very formula.

Gödel doesn’t say which statements may fall under this self-referential constraint, only that some such numbers are inevitable, numbers for which x is not simply a number but a Gödel number that can be decoded as the very statement itself. This final self-referential leap, that there is an arithmetical function that produces the Gödel number of his paradoxical statement, confirms that the statement is true, though its truth cannot be arrived at by a formal logical progression. Its provability is a quality of its being a correct arithmetical result, despite the impossibility of proving such a paradox through logic.

>> proving that something is impossible through a mathematical equation <<

Leaving the extraordinary method aside, we jump to the more straightforward implications. The first incompleteness theorem states that if a system of axioms is truly consistent, it will be incomplete: there will always be statements within the system that, while true, are not provable using only that system’s axioms. In these terms, for example, the difficulty of proving the arithmetical Goldbach conjecture may (perhaps) be an indication that it is a Gödelian “true, but unprovable” theorem (we still don’t know). The second incompleteness theorem is an extension of the first and states that any system that is in fact complete cannot prove its own consistency.

We can summarize this even more concisely: If any formal system that includes arithmetic is consistent, it is necessarily incomplete. And if such a system is actually complete, then it must be inconsistent. The treasured goal of simultaneous consistency and completeness was, from the beginning, an aspirational sham.

The primacy of mathematical logic and empirical science that the Vienna Circle had espoused was fundamentally shattered forever. If their ambition was to close the door on metaphysical thinking, Gödel blew that door right off its hinges. There are clearly gaps in understanding that will never be filled by scientists and logicians, gaps that only some forms of metaphysical intuitions could hope to fill.

[a-lesbian-next-door]

When you're having a jam session

And your family asks why you're catching an attitude??? Like you see the ear phones??

[one-angry-girl-archive-blog]

“If pornography made us healthy, we would be healthy by now. If the sex industry made us healthy, then those in the sex industry would be the healthiest and have the healthiest relationships.” -Dr. Mary Anne Layden

[the-crimson-stripe-blog]

Logic, like many other things, is a tool, by which we gain an understanding not only of us and our world, but of the other peoples around us and their worlds, as well as how the worlds relate to each other and their inhabitants. And so on and so forth.

Logic™ is the means by which a glorified confirmation bias is reinforced as truth, and all other aspects, perspectives, and qualities and everything else in existence is reductively judged in relation to that bias: does it validate the bias, refute the bias, or else ignore it?

And in humoring this method of thinking, the humanity we fight for is thus lost.

So, put shortly: Fuck Logic™.