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The Paradoxes Seriously Fuck Math
By Dr A. Crank

What an arrogant bunch of pricks, little ones, mathematicians are, me excluded (not arrogant; big prick). Let’s deflate these cock suckers some more.

The existence of logico-semantical paradoxes is a real problem for their paradigm. Mathematicians pride themselves on logical rigour beyond all things, but for over one hundred years, mathematics has faced foundational paradoxes. Before that, going back to ancient Greece, there was awareness of semantical paradoxes such as the Liar:

Here is a standard statement of the paradox:

“Consider a sentence named ‘FLiar’, which says of itself (i.e., says of FLiar) that it is false.

• FLiar:FLiar is false.

This seems to lead to contradiction as follows. If the sentence ‘FLiar is false’ is true, then given what it says, FLiar is false. But FLiar just is the sentence ‘FLiar is false’, so we can conclude that if FLiar is true, then FLiar is false. Conversely, if FLiar is false, then the sentence ‘FLiar is false’ is true. Again, FLiar just is the sentence ‘FLiar is false’, so we can conclude that if FLiar is false, then FLiar is true. We have thus shown that FLiar is false if and only if FLiar is true. But, now, if every sentence is true or false, FLiar itself is either true or false, in which case—given our reasoning above—it is both true and false. This is a contradiction. Contradictions, according to many logical theories (e.g., classical logic, intuitionistic logic, and much more) imply absurdity—triviality, that is, that every sentence is true.”

Fine, you may say, simply deny that all sentences are either true or false, the principle of bivalence. Many have done that, but there are strengthened Liar paradoxes that escape that method of defence:

“Consider a sentence named ‘ULiar’ (for ‘un-true’), which says of itself that it is not true.

• ULiar:ULiar is not true.

The argument towards contradiction is similar to the FLiar case. In short: if ULiar is true, then it is not true; and if it is not true, then it is true. But, now, if every sentence is true or not true, ULiar itself is true or not true, in which case it is both true and not true. This is a contradiction. According to many logical theories, a contradiction implies absurdity—triviality.”

The Liar paradox can also be presented in more complex forms that escape attempts to solve the strengthened versions, such as Yablo’s paradox, where there is use made of a list of sentences where reference is made to the sentence being not-true further down the list:

S1: S2: S3: Sn: For all m>1, Sm is false.

For all m>2, Sm is false.

For all m>3, Sm is false.

⋮ ⋮ ⋮ ⋮ ⋮

For all m>n, Sm is false.

⋮ ⋮ ⋮ ⋮ ⋮

This also generates a paradox.

In the Middle Ages logicians, who were also theologians, debated the meaning of various “insolubilia,” such as the Liar, but also other, perhaps more challenging paradoxes, such as the paradox of validity, attributed to “Pseudo-Scotus,” someone who was not John Duns Scotus, but whose work ended up in a publication by John Duns Scotus. Pseudo probably lived around the 1340s or 1350s. Here is a version of his paradox:

“Pseudo-Scotus does not realise he is dealing with a paradox. He presents it as an argument against a certain definition of valid consequence: that it is impossible for things to be as the premise signifies without being as the conclusion signifies. Consider the argument: God exists. So this argument is invalid. Call the argument, π. If π were valid, it would be a valid argument with a true premise, so the conclusion would be true, that is, π would be invalid. So by reductio, π is invalid. Now the conclusion of π is necessary, since we’ve just inferred it from the necessary truth that God exists. By the above definition, any argument with a necessarily true conclusion is valid. So by the definition an invalid argument is valid.

But π is, in fact, paradoxical. For by Pseudo-Scotus’ own admission, we’ve deduced the conclusion of π from its premise. First, we assumed π was valid. Then taking it that the premise of π was true, we inferred that its conclusion was true, that is, π was invalid. So by reductio, π is invalid, assuming that its premise is true. So on any account, π is valid, since its conclusion follows from its premise. Hence π is both valid and invalid … since π is valid, its conclusion is false, so its premise must be false too, that is, there is no God. Again, we could use this argument to disprove anything.”

The conclusion that there was no God was taken in the day as showing that the argument was absurd. A more modern version of this argument is as follows. Consider:

(A) 1=1


(B) This argument (i.e. (A) → (B)) is invalid.

Now, if this argument is valid, it has a true premise and a false conclusion, as every argument with a true premise(s) and false conclusion is invalid. Therefore, this argument is invalid. Therefore, the argument (A) → (B) is valid if it is invalid. So, by reductio ad absurdum, the argument is invalid. However, taking an alternative track, 1=1 is mathematically true, and hence a necessary truth. It is a principle of most modal logics that what deduced from a necessary true proposition is necessarily true. Then, as premise (B) is deduced from the necessary truth 1=1, then the argument (A) → (B) is valid. So, the conclusion (B) is a necessary truth, namely it is a necessary truth that the argument is invalid. Hence the argument is valid and not-valid, a contradiction!

There are even worse paradoxes that were uncovered in the 20th century, where by the fundamental principles of logic, one can prove any arbitrary proposition, p:

“Lob’s argument shows that the use of negation is not needed for the proof that every statement is true. For, let B be any sentence of the language. Create a sentence A such that A is true if and only if it implies B, i.e., (2) A if and only if (A→B). Then argue as follows. Suppose (3) A, then (4) A →B and (5) B. In other words, withdrawing the assumption (3), (6) A →B, i.e., (7) A, so (8) B!”

There are many responses to this triviality proof, but also many rejoinders restating the paradox. It seems that some fundamental logical principles must be given up, but what? Everything seems to be basic and undeniable.

I do not have a formal answer to give here, as I believe that one cannot ever be forthcoming. Every logician has a different solution, and these solutions all have specific defects. This to my warped mind shows that human reason is basically shit, limited in its capacity to understand the universe, and even to think, because we evolved to primarily drink our enemies blood from skulls, then die and rot.

The so-caled educated like to think that they have some divine right to pis on people like you. But, when it comes down to it, most of higher education is just bullshit. Here’s a book fullof all the stats about this, including the fact that most graduates basically remember nothing much of their courses after the exam:

I am one of the best-educated people on the planet, and I am here to tell you that most of what we learn, in every field is just bullshit, lies, fraud, deception:

“There is increasing concern that most current published research findings are false. The probability that a research claim is true may depend on study power and bias, the number of other studies on the same question, and, importantly, the ratio of true to no relationships among the relationships probed in each scientific field. In this framework, a research finding is less likely to be true when the studies conducted in a field are smaller; when effect sizes are smaller; when there is a greater number and lesser preselection of tested relationships; where there is greater flexibility in designs, definitions, outcomes, and analytical modes; when there is greater financial and other interest and prejudice; and when more teams are involved in a scientific field in chase of statistical significance. Simulations show that for most study designs and settings, it is more likely for a research claim to be false than true. Moreover, for many current scientific fields, claimed research findings may often be simply accurate measures of the prevailing bias. In this essay, I discuss the implications of these problems for the conduct and interpretation of research.”

Turd America

Trumpapocalypse Now: The Advent of an American Usurper at the fall of Western Civilization

Own the collected works of John Saxon, Professor X, Eirik Blood Axe, William Rapier and other counter culture critics, on Kindle, via the link below. Amazon:

The Great Train Wreck of the West

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