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God and Mathematics Part 1: The Logico-Semantical Paradoxes By Dr John Jensen
I am pleased to be holidaying in Australia taking a break from the dangers of living in the United States, which is now in a state of virtual civil war. Young Chris Knight, whom I knew from America as a student, invited me to write some papers in my spare time while here, which I am pleased to do. See, no rest for the good, as well as the wicked. My interest, as a Christian who believes in the literal truth of the Holy Scriptures is that science has been in many ways a threat to Biblical belief, in ways that are often subtle. Evolution is not subtle, nor is mechanistic accounts of the mind, which eliminate free will and the soul, but a subtle metaphysical threat is posed by certain so-called advances in mathematics, especially the theory of infinity, which in a sense deconstructs one of the divine attributes. If infinity is tamed and is just another mathematical object, as in the theory of transfinite set theory, then it is that much easier, along with the materialist baggage of the modern physical sciences, to abandon God. Who needs that hypothesis, Laplace said to Napoleon:
There is a current of thought which takes it that mathematics is the path to God, because God is Logos, and mathematics is allegedly Logos incarnate. Certainly, Jesus Christ, as a person of the Trinity, is an incarnation of the divine Logos that formed the universe, but it is drawing a long bow to therefore identify the divine Logos with an imperfect instrument such as mathematics. In these papers I explain why by illustrating some of the many theoretical limitations of logic and mathematics, and paradoxes and antinomies. This to my mind breaks the classical link which theologians have made with mathematics and the Divine.
Let’s begin with the logico-mathematical paradoxes, and examine their significance from a Christian perspective. Then, in other papers, I will move to attempting to refute some existing mathematical theories, as part of the necessary humbling effect. 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, but no doubt today, that would be celebrated by atheists. They should not celebrate for long.
A more modern version of this argument is as follows. Consider:
(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. I believe that this is a clear demonstration of the fallibility of human reason, due to original sin. From a Christian perspective, it jumps off the page at one. It leads to one recognising that one knows so little compared to God. But, for the materialist atheist, these sorts of results must be highly disturbing showing a great crack in their castles of glass: