Math was constructed from, and must, of necessity, consist of a series of operations. And consequently, all mathematics is reducible to a few simple operations. (Which is why computers can calculate.) In practice. everything we can think of can be reduced to adding or removing one, and the test of equality.
(As an aside, this is why we can explain more possibilities with mathematics than the physical universe can demonstrate in reality: because the universe does not have this level of freedom due to the apparent complexity of its interacting forces.)
The act of adding and subtracting the symbols we call numerals and positional numbers, is an obvious and common example of creating symbols to replace what would be tedious and incomprehensible repetitions.
This necessity to use symbols to condense information into usable components (categories) is what our brains need to do. Imagine trying to do all operations by counting? It would be impossible. We could not function without these symbols.
Furthermore, describing mathematical equations and proofs as operations is both verbally and syntactically burdensome. And since these operations are largely simple, and can be accurately reduced to symbols (named functions) there is little value in articulating them as operations.
So mathematicians have developed a multitude of symbols and names for what are not extant objects, but names of functions (sets of operations) – just as every other discipline creates heavily loaded terms in order to allow informationally dense communication with fewer words.
Most ‘numbers’ are anything but: they are names, glyphs and symbols, for functions that consist of large numbers of operations. “The natural numbers exist in nature, but all else is the work of man.”
The reason for this complexity is that quantitative, and directional relationships are expressed as ratios, and while some ratios are reducible to numbers, others are not. Those that are not reducible must be expressed as functions. We have not invented a mathematical system that can circumvent this problem. It is possible such a thing cannot be done.
Now aside from the practical utility of creating symbols, that obscure the operations, there is a practical value in using these names by disconnecting these names from their operations and from correspondence with any given scale.
That is, that disconnection allows one to use the logic of mathematics independent of cause, correspondence and scale, to explore ONLY the properties of the relations between the entities in question. And this turns out to be extremely useful for deducing what causes we do not now.
And this extraordinary utility has been responsible for the fact that the discipline has laundered time, causality and scale (precision) from the discipline. But one cannot say that a mathematical statement is true without correspondence with the real world. We can say it is internally consistent (a proof), but not that it is true (descriptive of reality via correspondence).
Mathematics when ‘wrong’ most recently, with Cantor’s sets, in which he used imaginary objects, infinity, the excluded middle and the the axiom of choice, to preserve this syntactical convenience of names, and in doing so, completed the diversion of mathematics from a logic of truth (external correspondence), to one that is merely a logic of proof (internal consistency).
Cantor’s work came at the expense of correspondence, and by consequence at the expense of truth. ie: mathematics does not determine truths, only proofs, because all correspondence has been removed by these ‘contrivances’, whose initial purpose was convenience, but whose accumulated errors have led to such (frankly, absurd) debates, .
So the problem with mathematical platonism, which turns out to be fairly useful for the convenience of practitioners, is not so much a technical problem but a MORAL ONE. First, mathematicians, even the best, rarely grasp this concept. Second, since, because it is EASIER to construct mathematical proofs than any other form of logic, it is the gold standard for other forms of logic. And the envy of other disciplines. And as such mathematical platonism has ‘bled’ into other envious fields, the same way that Physics has bled into economics.
Worse, this multi-axial new mysticism has been adopted by philosophers from Kant to the Frankfurt school to the postmodernists, to contemporary totalitarian humanists as a vehicle for reinserting arational mysticism into political debate – as a means of obtaining power.
Quite contrary to academic opinion, all totalitarianism is, is catholicism restated in non-religious terms, with the academy replacing the church as the constructor of obscurant language.
I suspect this fairly significant error is what has plagued the physics community, but we have found no alternative to current approaches. Albeit, I expect, that if we retrained mathematicians, physicists, and economists to require operational language in the expression of mathematical relations, that whatever error we are making in our understanding of physics would emerge within a generation.
No infinity can exist. Because no operation can be performed infinitely. We can however, adjust the precision and scale of any proof to suit the context, since any mathematical expression, consists of ratios that, if correspond to reality, we can arbitrarily adjust for increasing precision.
Mathematics cannot claim truth without correspondence.
Correspondence in measures is a function of scale and the UTILITY of precision, in the CONTEXT of which the operation is calculated (limit).
A language of mathematics that is described independent of scale in given context, can be correctly stated. It need not be magian.
Fields can still be understood to be imaginary patterns.
But the entire reason that we find such things interesting, is a folly of the mind, no different from the illusion of movement in a film.
The real world exists. We are weak computers of property in pursuit of our reproduction and amusement. We developed many forms of instrumentalism to extend our weak abilities. We must use instruments and methods to reduce to analogies to experience, those things which we cannot directly do so.
It’s just that simple.
AGAINST THE PLATONIC (IMAGINARY) WORLD
Why must we support imaginary objects, as extant? Especially when the constructive argument (intuitionist) in operational language, can provide equal explanatory power?
Why must we rely on ZFC+AC when we have recursive math, or when we can explain all mathematics in operational language without loss of context, scale, precision and utility? Just ’cause it’s easier.
But that complexity is a defense against obscurantism and platonism. So it is merely a matter of cost.
I understand Popper as trying to solve a problem of meta ethics, rather than anything particularly scientific. And I see most of his work as doing the best he could for the purposes that I’ve stated.
Anyone who disagrees with me would have to disagree with my premies and my argument, not rely on the existence of platonist entities (magic) in order to win such an argument.
That this is impossible, is at least something that I understand if no one else yet does. I don’t so much need someone to agree with me as constantly improve my argument so that I can test and harden it until it is unassailable or defeated.
I think that defeating this argument is going to be very, very, difficult.
TIME AND OPERATIONS (ACTIONS) IN TIME
One cannot state that abstract ideas can be constructed independent of time, or even that they could be identified without changes in state over time. Or that thought can occur without the passage of time. Or consciousness can occur without the passage of time.
Whether I make one choice or another is not material. This question is not a matter of choice, it is a matter of possibility. I can make no choice without the passage of time.
I think that the only certain knowledge consists of negations, and that all the rest is conjecture. This is the only moral position to take. And it is the only moral position since argument exists for the purpose of persuasion, and persuasion for cooperation.
I keep seeing this sort of desire to promote the rather obvious idea that induction is nonsense – yet everyone uses it, as a tremendous diversion from the fact that induction is necessary for action in real time, whenever the cost of not acting is higher than the cost of acting.
Description, deduction, induction, abduction, guessing and intuitive choice are just descriptions of the processes we must use given the amount of information at our disposal. Science has no urgency, and life threatening emergencies do.
Popper (and CR-ists for that matter) seem to want to perpetuate either mysticism, or skepticism as religion, rather than make the very simple point that the demands for ‘truth’ increase and decrease given the necessity of acting in time.
I guess that I could take a psychological detour into why people would want to do this. But I suspect that I am correct (as I stated in one of these posts) that popper was, as part of his era, trying to react against the use of science and academia to replace the coercive power of the church. So he restated skepticism by establishing very high criteria for scientific truth.
And all the nonsense that continues to be written about his work seek to read into platonic tea leaves, when the facts are quite SIMPLE. (Back to Argumentation Ethics at this point.) The fact is that humans must act in real time and as the urgency of action increases so does the demand for truth. Conversely, as the demand for cooperation increases, the demand for truth increases. Finally at the top of the scale we have science, which in itself is an expensive pursuit, and as such one is forbidden to externalize costs to other scientists. (Although if we look at papers this doesn’t actually work that well except at the very top margin.)
THE QUESTION IS ONE OF COOPERATION
The problem is ECONOMIC AND COOPERATIVE AND MORAL, not scientific.
It’s just that simple. We cannot disconnect argument from cooperation without entering the platonic. We cannot disconnect math from context without entering the platonic. We cannot disconnect numbers from identity without entering the platonic.
Each form of logic constrains the other. But the logic that constrains them all, is action. Without action, we end up with the delusions we spend most of philosophical discourse on. It’s all nonsense.
I understand the difference between the real and the unreal, and the necessity of our various logics as instruments for the reduction of that which we cannot comprehend (sympathize with) to analogies to experience that we can comprehend ( sympathize with).
Which is profound if you grasp it.
THE PROBLEM OF SYMBOLS AND ECONOMY OF LANGUAGE
If you cannot describe something as human action, then you do not understand it. Operational language is the most important, and least articulated canon of science.
I do not argue against the economy of language. I argue against the loss of causality and correspondence that accompanies repeated use of economizing terms.
( I am pretty sure I put a bullet in this topic along with apriorism in economics. )
MORAL STANDARDS OF TRUTH
Requiring a higher standard of truth places a higher barrier on cooperation.
This is most important in matters of involuntary transfer, such as taxation or social and moral norms.
Religions place an impossible standard of truth. This is why they are used so effectively to resist the state. Religious doctrine reliant upon faith is argumentatively inviolable.
As such, no cooperation can be asked or offered outside of their established terms. … It’s brilliant really. Its why religious groups can resist the predation of the state.
I would prefer instead we relied upon a prohibition on obscurant language and the requisite illustration of involuntary transfers, such that exchanges were easily made possible, and discounts (thefts) made nearly impossible.
This is, the correct criteria for CR, not the platonic one that is assumed. In this light CR looks correct in practice if incorrect in argument.
(There. I did it. Took me a bit.)