|, ||, |||, ||||| . The Non-Metaphysics of Unprediction. Thought of the day 67.1

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The cornerstone of Hilbert’s philosophy of mathematics was the so-called finitary standpoint. This methodological standpoint consists in a restriction of mathematical thought to objects which are “intuitively present as immediate experience prior to all thought,” and to those operations on and methods of reasoning about such objects which do not require the introduction of abstract concepts, in particular, require no appeal to completed infinite totalities.

Hilbert characterized the domain of finitary reasoning in a well-known paragraph:

[A]s a condition for the use of logical inferences and the performance of logical operations, something must already be given to our faculty of representation, certain extra-logical concrete objects that are intuitively present as immediate experience prior to all thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that can neither be reduced to anything else nor requires reduction. This is the basic philosophical position that I consider requisite for mathematics and, in general, for all scientific thinking, understanding, and communication. [Hilbert in German + DJVU link here in English]

These objects are, for Hilbert, the signs. For the domain of contentual number theory, the signs in question are sequences of strokes (“numerals”) such as

|, ||, |||, ||||| .

The question of how exactly Hilbert understood the numerals is difficult to answer. What is clear in any case is that they are logically primitive, i.e., they are neither concepts (as Frege’s numbers are) nor sets. For Hilbert, the important issue is not primarily their metaphysical status (abstract versus concrete in the current sense of these terms), but that they do not enter into logical relations, e.g., they cannot be predicated of anything.

Sometimes Hilbert’s view is presented as if Hilbert claimed that the numbers are signs on paper. It is important to stress that this is a misrepresentation, that the numerals are not physical objects in the sense that truths of elementary number theory are dependent only on external physical facts or even physical possibilities. Hilbert made too much of the fact that for all we know, neither the infinitely small nor the infinitely large are actualized in physical space and time, yet he certainly held that the number of strokes in a numeral is at least potentially infinite. It is also essential to the conception that the numerals are sequences of one kind of sign, and that they are somehow dependent on being grasped as such a sequence, that they do not exist independently of our intuition of them. Only our seeing or using “||||” as a sequence of 4 strokes as opposed to a sequence of 2 symbols of the form “||” makes “||||” into the numeral that it is. This raises the question of individuation of stroke symbols. An alternative account would have numerals be mental constructions. According to Hilber, the numerals are given in our representation, but they are not merely subjective “mental cartoons”.

One version of this view would be to hold that the numerals are types of stroke-symbols as represented in intuition. At first glance, this seems to be a viable reading of Hilbert. It takes care of the difficulties that the reading of numerals-as-tokens (both physical and mental) faces, and it gives an account of how numerals can be dependent on their intuitive construction while at the same time not being created by thought.

Types are ordinarily considered to be abstract objects and not located in space or time. Taking the numerals as intuitive representations of sign types might commit us to taking these abstract objects as existing independently of their intuitive representation. That numerals are “space- and timeless” is a consequence that already thought could be drawn from Hilbert’s statements. The reason is that a view on which numerals are space- and timeless objects existing independently of us would be committed to them existing simultaneously as a completed totality, and this is exactly what Hilbert is objecting to.

It is by no means compatible, however, with Hilbert’s basic thoughts to introduce the numbers as ideal objects “with quite different determinations from those of sensible objects,” “which exist entirely independent of us.” By this we would go beyond the domain of the immediately certain. In particular, this would be evident in the fact that we would consequently have to assume the numbers as all existing simultaneously. But this would mean to assume at the outset that which Hilbert considers to be problematic.  Another open question in this regard is exactly what Hilbert meant by “concrete.” He very likely did not use it in the same sense as it is used today, i.e., as characteristic of spatio-temporal physical objects in contrast to “abstract” objects. However, sign types certainly are different from full-fledged abstracta like pure sets in that all their tokens are concrete.

Now what is the epistemological status of the finitary objects? In order to carry out the task of providing a secure foundation for infinitary mathematics, access to finitary objects must be immediate and certain. Hilbert’s philosophical background was broadly Kantian. Hilbert’s characterization of finitism often refers to Kantian intuition, and the objects of finitism as objects given intuitively. Indeed, in Kant’s epistemology, immediacy is a defining characteristic of intuitive knowledge. The question is, what kind of intuition is at play? Whereas the intuition involved in Hilbert’s early papers was a kind of perceptual intuition, in later writings it is identified as a form of pure intuition in the Kantian sense. Hilbert later sees the finite mode of thought as a separate source of a priori knowledge in addition to pure intuition (e.g., of space) and reason, claiming that he has “recognized and characterized the third source of knowledge that accompanies experience and logic.” Hilbert justifies finitary knowledge in broadly Kantian terms (without however going so far as to provide a transcendental deduction), characterizing finitary reasoning as the kind of reasoning that underlies all mathematical, and indeed, scientific, thinking, and without which such thought would be impossible.

The simplest finitary propositions are those about equality and inequality of numerals. The finite standpoint moreover allows operations on finitary objects. Here the most basic is that of concatenation. The concatenation of the numerals || and ||| is communicated as “2 + 3,” and the statement that || concatenated with ||| results in the same numeral as ||| concatenated with || by “2 + 3 = 3 + 2.” In actual proof-theoretic practice, as well as explicitly, these basic operations are generalized to operations defined by recursion, paradigmatically, primitive recursion, e.g., multiplication and exponentiation. Roughly, a primitive recursive definition of a numerical operation is one in which the function to be defined, f , is given by two equations

f(0, m) = g(m)

f(n′, m) = h(n, m, f(n, m)),

where g and h are functions already defined, and n′ is the successor numeral to n. For instance, if we accept the function g(m) = m (the constant function) and h(n, m, k) = m + k as finitary, then the equations above define a finitary function, in this case, multiplication f (n, m) = n × m. Similarly, finitary judgments may involve not just equality or inequality but also basic decidable properties, such as “is a prime.” This is finitarily acceptable as long as the characteristic function of such a property is itself finitary: For instance, the operation which transforms a numeral to | if it is prime and to || otherwise can be defined by primitive recursion and is hence finitary. Such finitary propositions may be combined by the usual logical operations of conjunction, disjunction, negation, but also bounded quantification. The problematic finitary propositions are those that express general facts about numerals such as that 1 + n = n + 1 for any given numeral n. It is problematic because, for Hilbert it is from the finitist point of view incapable of being negated. By this he means that the contradictory proposition that there is a numeral n for which 1 + n ≠ n + 1 is not finitarily meaningful. A finitary general proposition is not to be understood as an infinite conjunction but only as a hypothetical judgment that comes to assert something when a numeral is given. Even though they are problematic in this sense, general finitary statements are of particular importance to Hilbert’s proof theory, since the statement of consistency of a formal system T is of such a general form: for any given sequence p of formulas, p is not a derivation of a contradiction in T. Even though in general existential statements are not finitarily meaningful, they may be given finitary meaning if the witness is given by a finitary function. For instance, the finitary content of Euclid’s theorem that for every prime p there is a prime > p, is that given a specific prime p one can produce, by a finitary operation, another prime > p (viz., by testing all numbers between p and p! + 1.).

ε-calculus and Hilbert’s Contentual Number Theory: Proselytizing Intuitionism. Thought of the Day 67.0

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Hilbert came to reject Russell’s logicist solution to the consistency problem for arithmetic, mainly for the reason that the axiom of reducibility cannot be accepted as a purely logical axiom. He concluded that the aim of reducing set theory, and with it the usual methods of analysis, to logic, has not been achieved today and maybe cannot be achieved at all. At the same time, Brouwer’s intuitionist mathematics gained currency. In particular, Hilbert’s former student Hermann Weyl converted to intuitionism.

According to Hilbert, there is a privileged part of mathematics, contentual elementary number theory, which relies only on a “purely intuitive basis of concrete signs.” Whereas the operating with abstract concepts was considered “inadequate and uncertain,” there is a realm of extra-logical discrete objects, which exist intuitively as immediate experience before all thought. If logical inference is to be certain, then these objects must be capable of being completely surveyed in all their parts, and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something which cannot be reduced to something else.

The objects in questions are signs, both numerals and the signs that make up formulas a formal proofs. The domain of contentual number theory consists in the finitary numerals, i.e., sequences of strokes. These have no meaning, i.e., they do not stand for abstract objects, but they can be operated on (e.g., concatenated) and compared. Knowledge of their properties and relations is intuitive and unmediated by logical inference. Contentual number theory developed this way is secure, according to Hilbert: no contradictions can arise simply because there is no logical structure in the propositions of contentual number theory. The intuitive-contentual operations with signs form the basis of Hilbert’s meta-mathematics. Just as contentual number theory operates with sequences of strokes, so meta-mathematics operates with sequences of symbols (formulas, proofs). Formulas and proofs can be syntactically manipulated, and the properties and relationships of formulas and proofs are similarly based in a logic-free intuitive capacity which guarantees certainty of knowledge about formulas and proofs arrived at by such syntactic operations. Mathematics itself, however, operates with abstract concepts, e.g., quantifiers, sets, functions, and uses logical inference based on principles such as mathematical induction or the principle of the excluded middle. These “concept-formations” and modes of reasoning had been criticized by Brouwer and others on grounds that they presuppose infinite totalities as given, or that they involve impredicative definitions. Hilbert’s aim was to justify their use. To this end, he pointed out that they can be formalized in axiomatic systems (such as that of Principia or those developed by Hilbert himself), and mathematical propositions and proofs thus turn into formulas and derivations from axioms according to strictly circumscribed rules of derivation. Mathematics, to Hilbert, “becomes an inventory of provable formulas.” In this way the proofs of mathematics are subject to metamathematical, contentual investigation. The goal of Hilbert is then to give a contentual, meta-mathematical proof that there can be no derivation of a contradiction, i.e., no formal derivation of a formula A and of its negation ¬A.

Hilbert and Bernays developed the ε-calculus as their definitive formalism for axiom systems for arithmetic and analysis, and the so-called ε-substitution method as the preferred approach to giving consistency proofs. Briefly, the ε-calculus is a formalism that includes ε as a term-forming operator. If A(x) is a formula, then εxA(x) is a term, which intuitively stands for a witness for A(x). In a logical formalism containing the ε-operator, the quantifiers can be defined by: ∃x A(x) ≡ A(εxA(x)) and ∀x A(x) ≡ A(εx¬A(x)). The only additional axiom necessary is the so-called “transfinite axiom,” A(t) → A(εxA(x)). Based on this idea, Hilbert and his collaborators developed axiomatizations of number theory and analysis. Consistency proofs for these systems were then given using the ε-substitution method. The idea of this method is, roughly, that the ε-terms εxA(x) occurring in a formal proof are replaced by actual numerals, resulting in a quantifier-free proof. Suppose we had a (suitably normalized) derivation of 0 = 1 that contains only one ε-term εxA(x). Replace all occurrences of εxA(x) by 0. The instances of the transfinite axiom then are all of the form A(t) → A(0). Since no other ε-terms occur in the proof, A(t) and A(0) are basic numerical formulas without quantifiers and, we may assume, also without free variables. So they can be evaluated by finitary calculation. If all such instances turn out to be true numerical formulas, we are done. If not, this must be because A(t) is true for some t, and A(0) is false. Then replace εxA(x) instead by n, where n is the numerical value of the term t. The resulting proof is then seen to be a derivation of 0 = 1 from true, purely numerical formulas using only modus ponens, and this is impossible. Indeed, the procedure works with only slight modifications even in the presence of the induction axiom, which in the ε-calculus takes the form of a least number principle: A(t) → εxA(x) ≤ t, which intuitively requires εxA(x) to be the least witness for A(x).

The Feedback of Capital and Standard of Living. Some Wayside Didactics.

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It is often said to countries in trouble that their people were living above their standards. That their consumption is higher than their production. This, in fact, is true … for everybody on this planet. In financial terms.

Look at the image of Figure 1. People (Labour Power), together with machines from the capital (MoP) produce goods that only (mostly) humans consume. Left the production, right the consumption.

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Figure 1: Production and consumption of humans and capital

If everything that is produced is consumed (according to Jean Baptiste Say), it is obvious that humans consume more than they produce. This seems contradictory with the ideas of Marx, but it isn’t. Marx said that Labour Power with the help of MoP produces, and that this production is fully attributed to Labour Power and is thus skimmed when it consumes less than this production. We can also equally well say that MoP (‘capital’) is producing with the help of Labour Power. Or just say that both are producing and say that each is the right ‘owner’ of its own production.

In the above figure, the arrows show the flow of production-consumption. The payment for produced products is an arrow in opposite direction. In this example, humans get 95% of consumption while they do only 50% of the production. They thus also only get 50% of payment. The rest of the consumption is paid by ‘borrowing’ money somehow, and they live above their standard. The payment goes 50% to the capital. But, because capital does not consume, this payment is used to increase the capital. Two extreme scenarios:

• The money for payment of production is fully in the form of a loan to the humans. Money starts thus accumulating at the capital.

• The money for payment is fully used to invest in new capital. In that case, the ‘consumption’ of capital is 50%, but after one cycle, a larger part of the production is done by capital. See figure 2 below.

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Figure 2: Production and consumption of humans and capital, if the capital consumes as human, but this consumption is used as new starting capital in a new cycle

In the first step, 50% of the production and consumption is done by capital. In the second cycle it is already 67%. In the third cycle it is 80%, then 89%, etc. In general 2n−1/(2n−1 + 1) at step n; capital doubles at every cycle, where humans stay constant. The final situation is that 100% of production is done by capital. Obviously, sooner or later the system has to switch to the first scenario.

In either scenario, the capital accumulates. The basic ingredient is that capital does not need consumption for its survival; any ‘consumption’ is directly converted into more capital. The system will probably have a mix of the two. After all, capital cannot go on doubling all the time.

So, we see that capital is condensing at the capital. That is because the means of production – other than human labor – do not consume, and, therefore, humans do consume more than they produce, and the means of production (machines) do consume less than they produce, with the total in a zero-sum-game way consuming exactly what they produce. The owners of the means of production get the rights to consumption and these rights are constantly increasing. It is a positive-feedback run-away system.

Let’s put this in an example to explain it better. Imagine I make clothespins and so does my neighbor. However, my neighbor has slightly more costs than me, or is slightly less productive for some reason (work accident, or so). He earns just enough to survive. He makes one ’unit’ and this barely covers the cost of life, which is also minimally 1 unit. I am slightly more productive, or my cost of living is slightly lower. Therefore, I can save a little ‘money’. Let’s assume the former, I am more productive. Now, either I make 1.1 units and the surplus 0.1 units I trade for a clothespin machine, or I work a little less on making clothespins and in this spare time – one hour per day – I make the machine myself. Let’s assume the second scenario, because it is easier reasoning, although they are equivalent. We both make two ‘units’ of pins, sell them and buy things (two units worth) to survive. I however, make as well a machine that makes pins.

After finishing my machine, maybe after ten years, the total production goes up. The demand for our pins stays the same. The markets needs two units of clothespins. It now means that I will get more share of the profit. Imagine my machine makes as much units as a human can, one unit per year. We thus have three units to offer to the market. The price of pins on the market could (and will) drop through the mechanism of supply and demand. In principle down to 67% of the original price. Not lower, because that would imply that the total price of more pins would be lower than before.

To make it simple, imagine exactly that happens. The price is 2/3; one unit of pins gives only 2/3 consumption rights. We sell three units and thus get a total of two units of consumption rights. These are distributed over the production units. My neighbor has one third of the production units and thus gets 1/3 share of the consumption rights, a total of 2/3 units. I and my machine get 2/3 share, 4/3 consumption rights. Note that I confiscate – skim – the production rights of my ‘slave’ machine.

Now my neighbor has a problem. He gets 2/3 units of consumption rights, there where one full unit is needed to survive. He did not start working less, or become less productive, or lazy. He simply lost his percentage share of the means of production. And once this starts, there is no stopping it. It in fact accelerates.

There are two scenarios. Either I keep producing pins myself, as shown above, resulting in immediate misery for my neighbor, or I stop working altogether on making pins manually, and we go back to the situation where we make two units of pins, sell them, and each one gets one unit of consumption rights. However, now I have 100% free time (my machine doing all the work), and I can dedicate it to make a new machine. This takes only one year instead of ten, since I now have 100% free time, instead of only 10%. In the first situation, I could lend 1/3 of my consumption rights to my neighbor. However – nothing is for free in this life – next year I want 10% profit on my loan. His problems will be bigger next year. Next year I will refinance his loan. Etc. The reader will easily understand that my neighbor will wind up being my feudal possession. I will take everything he owns. Instead, I could opt for the second path, producing a new machine in my spare time. In that case, next year we will have 4 production units, my neighbor and I as human labor, and two mechanical units. These mechanical units are mine and will claim the consumption rights; together with my own labor, I will now get 75% of the two consumption rights. 1.5 for me and 0.5 for my neighbor. This path leads to the state where I have 100% of the consumption rights. Or I can again decide to use part or all of my human labor or machine power to make new machinery. Sooner or later, anyway, my neighbor will have to borrow consumption rights from me. This is a feedback system. Any small perturbation results in a saturation in which I will get 100% of the consumption rights and where I will wind up being the feudal lord of my neighbor. One could argue that this reasoning does not work, because the rest of the world is also increasing productivity and the price of the products offered by them (and the cost of living for me and my neighbor) goes down, as fast as the price of our clothespins go down and we will both easily survive. First of all, we consider here only the local effect, independent of the full market. Technological innovation creates immediate misery for some, a deterioration of life while these people are doing nothing worse. Second, when the rest of the market is behaving in the same way, we remain with an overall effect of condensation of wealth. Capital attracts capital. This is a form of the Matthew Effect, named after the apostle from the bible, transferring money from the poor to the rich. Matthew 25:29,

For onto everyone that hath shall be given, and he shall have abundance, but from him that hath not shall be taken away even that which he hath.