The initial optimism: Rationalists see the beauty of mathematics as its freedom from opinion. Calculation yields one answer – monotruth! Like: 1 + 1 = 2! Well, it used to!
Unaided “human-independent” knowledge has come a long way since Pythagoras proclaimed that “all is number.” Plato acknowledged this but soon preferred geometry. Both thinkers implied that matter was incarnate mathematics, thereby linking abstract mathematics to a deterministic view of reality. Plato’s famous pupil, Aristotle, wisely dismissed the “human-independent” understandings of the mathematical calming cloud as opinion.
In the book, Polyscience and Christianity, the chapter on Old Mathematics; Old Sciences; Old Biases begins the review of the fascinating history of science.
Did you know that number initially began with one, arithmetic began with the number two and geometry with three? (Mathematics then began a journey towards complexity.) Then the Pythagoreans found the scandalous irrational numbers. Thinkers also quickly found that there was no zero – nature abhors a vacuum. Nor were there negative or fractional numbers – matter could not be fractured. Then some introduced infinity.
Later, there were additions to the mathematical operations of addition, subtraction, multiplication and division. Trigonometry used the scandalous irrational numbers. Algebra dabbled with calculating square and cubic roots of numbers – and discovered multiple answers! All answers are mathematically correct but are they physically correct?
A x B is the same as B x A, except when not allowed! Even later, there were “new” numbers – even multiple types of infinities.
What did all of these mean to monotruth?
The loss of monotruth: In recent centuries, thinkers reluctantly realized that mathematics did not always yield monotruth! Plato’s universal Mathematical certainty is a myth! Further, the intermediate steps of advanced calculations are “hard” to visualize as actually occurring in physical objects. The calculations are, at best, “snapshots” of thin slices of reality.
Geometries proliferate. Number theory is down to “only” four foundational schools! One school proclaims that number is based on infinite sets of empty sets, (set-theoretic method) – this is the “new math” taught to our children.
Rationalists cannot endow Mathematics with objectivity, it is the domain of Kurt Gӧedel’s “incompleteness theorem”. Truth therein is fuzzy. Consensus of learned participants provides false comfort.
What is the connection between mathematics and reality?
Mathematics demoted to an abstraction: Modern learned discussions ignore this loss of an “incarnate” link between mathematics and reality. Moderns can only hope for internal coherence and to save (some of) the appearances. Frankly, mathematics is an analytical assistant to science’s destructive interrogation; it, itself, is not science.
Is Plato’s Mathematical vision an example of Galileo’s “worlds on paper!” Many now say the truthfulness of complex mathematics can be resurrected by demonstrating many intersections with reality. This only occurs in the realm of abstractions and the otherwise inaccessible, but not in the domain of theory-free knowledge!
Rational thought is no longer reasonable. It has slid from causality back to wordy saving (some of) the appearances.
Connecting to universal knowledge; a new optimism(?): Nothing can impede the calming cloud of Mathematical determinism – until we remember that it is an unreal abstraction; it can be a word fog. Empiricism reminds us that human-sourced objective “truth” does not extend beyond theory-free knowledge, which, in turn, is comfortable with the physical extrapolative boundaries set by ex nihilo creation. This limit is a finger-print of theism.
From the previous blog, Plato’s writing allows Mathematics to be the great ahistorical leveler, which dissolves events (discontinuity) into myth. Yet this attitude also turns events such as earthquakes, volcanos and tsunamis into myth. They have to be ad hoc accounted for. But once they are added, the ahistorical ideal has vanished! Thinkers are then limited to a much reduced extrapolation – to the playing field and recorded history.
So, no wonder ex nihilo creation is such a show stopper. Mini or mega events disqualify the big picture presented by Mathematical determinism. Events are a stumbling block to Plato’s vision of the Mathematical calming cloud.
Deists – with their quiescent (absentee landlord) god – also see no events or boundaries. Deists join with Mathematical determinists, who start with atheism. Both shared an enthusiasm for infinity, but both were ambushed by atheorism!
Realistic optimism! Yes, mathematics can be unruly. Oh! For the good old event-filled days of theory-free knowledge and natural numbers!