(advanced)
—“I’ve been following your work for some time. I’m going through your reading list and am intrigued. One thing however was bugging me for some time. Namely, your attack on Cantor you gave on one of your interviews.
I’d greatly appreciate if you could expand on this a little. For example infinities of different sizes were very helpful in answering a very practical question of decidability of the halting problem.Could you give some very brief explanation?”—
[M]any analogies and models are useful, but are not true. Fairy tales are useful, and may in fact be the ultimate form of pedagogy but they are not true. Lying is profoundly useful. Propaganda is perhaps the most useful of technologies.
I will see if I can do this topic justice. I am not sure I can do it briefly. So I will give a few hints and see if you can make the connections rather than write five pages of text that I don’t have time for right now.
Lets understand that my criticism is an attempt to require mathematicians to practice their work ethically and morally and free of externality. And that as a cosmopolitan, I criticize cantor for unscientific method of argument that produces ‘meantinful’ but ‘untrue’ externalities, in a case where scientific statements that are equally meaningful but produce no untrue externalities will suffice. I am particularly concerned about this for the reasons the Intuitionists were concerned: Einstein should not have been revolutionary, and should have occurred a century earlier. And for the same reasons scientists publish in operational definitions and postmodernist pseudoscientists publish in ‘meaning and allegory’ – non-operational statements. Becuase there is a very great difference between a Name of something extant, and an Analogy to Experience. The former is laundered of imaginary content and the latter loaded with it. Or more precisely, the former is more true and the latter almost always false.
Cantor’s insight would be trivial if we taught the foundations of mathematics to children instead of taught by wrote memorization. The foundation being ‘pairing off’. Mathematics evolved from the very simple act of putting stone in a bag for every sheep one took out to the pasture at night, and one out of the bag for every sheep that one brought in. This is ‘pairing off’. Cantor returns us to the basis of mathematics by reminding us that we are at all times, paring off. And that we can pair off different bags of stones as well. We can also create a bag that in theory will always have more stones in it because in practice we can always find more stones on the beaches with which to refill the bag. We can use stones of different colors, sizes and textures. We can also name stones. But humans can only remember so many names so we invented positional naming: what we call ‘numbers’, consists of a sequence of operations by which we generate names, each of which is unique and whose name is positionally commensurable with all other names of stones regardless of size, texture and color.
The point I make here is that mathematics consists of sequences of operations, all of which use pairing off (category), positional naming(identity), and functions (collections of operations) to express ratios. All of which are existentially possible operations, that because of ‘pairing off’ correspond to the real world.
We can however, construct general rules of arbitrary precision by ignoring correspondence with any real world entity and instead comparing ratios of names against names. This arbitrary precision however eliminates contextual decidability. We now must construct a what we call a ‘limit’ for any ratio to be decidable. This limit corresponds to a real-world context.
For example, the square of two cannot logically exist without an expressed limit to the number of operations that must be performed. Yet neither can one perform an unlimited number of operations. So we have a general logical rule, not a number, because that number is existentially impossible to exist other than as a function decidable by contextual limit (limit of arbitrary precision).
Furthermore, we can use symbols to form recipes for these operations, and additional symbols for functions (collections of operations into a recipe). In this sense only natural numbers scientifically exist. All other ‘numbers’ that we refer to are existentially and necessarily, irrefutably, names of functions, not in fact numbers. We can use these functions as we use numbers, but they remain functions at all times out of existential necessity. Applying the name ‘number’ to a ‘function’ is a verbal convenience, like so many verbal conveniences in mathematics. But it is not ‘true’. This is the most common pseudoscientific fallacy in mathematics, and has been understood for over a century.
Religious mysticism works. Mathematical Platonism ‘works’. Both have the same scientific standing: pseudoscience or utter falsehood. We criticize the externalities of religious mysticism. I criticize the externalities of philosophical rationalism. Mathematicians of great skill still talk in terms of a non-existent mathematical reality instead of ‘the deterministic consequences of an axiomatic definition that appears to the human mind real because we are unable to imagine those relations as entirely deterministic.”
So let us look at infinity. Can any infinity exist? Well no extant infinity can exist, because there is nothing infinite that we can identify, and anythign we construct logically as infinite (a path around a circle) is limited by the boundaries of the universe, or limited by the number of operations we perform….. OR….. ***limited by the rate of operations we perform***.
What Cantor’s ‘analogy’ does, is imagine that all operations are performed instantaneously, and that the rate of one set of operations is faster than another rate of operations. In other words, he’s using the time honored principle of GEARS.
Now, is one infinity bigger than another? No. One set of operations produces more outputs per cycle of operations than another set of operations. One rate is faster than another rate. If we ignore the passage of time, then in any system the rate of production no matter how long will produce more operations in one than the other.
But, just as length did not exist as the constant, as Einstein showed us, neither do rates, also as Einstein showed us. Lengths are externally dependent on the observer as are rates.
Now, can any infinity exist? No. No infinity can exist. Infinity cannot exist any more than the square root of two can exist. Infinity is a name for a limit of arbitrary precision: information provided external to the calculation, useful when we wish to construct a scale independent general rule.
So let me play economist here, and ask the question “what is the total cost of mathematical platonism and the ignorance of mathematicians of the very simple fact that much of their language is pseudoscience justified by special pleading?” The answer I suspect, is that mathematics is quite simple and most people are limited in the application of it and access to it, simply because it remains taught to the general public as an ancient form of mysticism, rather than a very basic principle: bags, stones, and moving them around.
What has been the impact on physical science and mathematics? I am not sure. What has been the impact on the perpetuation of pseudoscience in the public mind: that appears to be vast.
Half truths are a pretty serious problem as precision increases. This is the direction of man’s evolution: toward greater truth. And greater truth means greater parsimony: greater precision. And greater precision means greater correspondence. We can know names rather than analogies. When we speak in the language of truth, using the true names of the universe, we will indeed be gods of it.
And mathematical platonism is for a variety of reasons one of the means by which modern pseudoscience in all walks of life has been perpetuated.
Curt Doolittle
The Philosophy of Aristocracy
The Propertarian Institute, Kiev, Ukraine