Highest Reality. Thought of the Day 70.0

Spiritual-awakening-higher-consciousness

यावचिन्त्यावात्मास्य शक्तिश्चैतौ परमार्थो भवतः॥१॥

Yāvacintyāvātmāsya śaktiścaitau paramārtho bhavataḥ

These two (etau), the Self (ātmā) and (ca) His (asya) Power (śaktiḥ) —who (yau) (are) inconceivable (acintyau)—, constitute (bhavataḥ) the Highest Reality (parama-arthaḥ)

The Self is the Core of all, and His Power has become all. I call the Core “the Self” for the sake of bringing more light instead of more darkness. If I had called Him “Śiva”, some people might consider Him as the well-known puranic Śiva who is a great ascetic living in a cave and whose main task consists in destroying the universe, etc. Other people would think that, as Viṣṇu is greater than Śiva, he should be the Core of all and not Śiva. In turn, there is also a tendency to regard Śiva like impersonal while Viṣṇu is personal. There is no end to spiritual foolishness indeed, because there is no difference between Śiva and Viṣṇu really. Anyway, other people could suggest that a better name would be Brahman, etc. In order not to fall into all that ignorant mess of names and viewpoints, I chose to assign the name “Self” to the Core of all. In the end, when spiritual enlightenment arrives, one’s own mind is withdrawn (as I will tell by an aphorism later on), and consequently there is nobody to think about if “This Core of all” is personal, impersonal, Śiva, Viṣṇu, Brahman, etc. Ego just collapses and This that remains is the Self as He essentially is.

He and His Power are completely inconceivable, i.e. beyond the mental sphere. The Play of names, viewpoints and such is performed by His Power, which is always so frisky. All in all, the constant question is always: “Is oneself completely free like the Self?”. If the answer is “Yes”, one has accomplished the goal of life. And if the answer is “No”, one must get rid of his own bondage somehow then. The Self and His Power constitute the Highest Reality. Once you can attain them, so to speak, you are completely free like Them both. The Self and His Power are “two” only in the sphere of words, because as a matter of fact they form one compact mass of Absolute Freedom and Bliss. Just as the sun can be divided into “core of the sun, surface of the sun, crown”, etc.

तयोरुभयोः स्वरूपं स्वातन्त्र्यानन्दात्मकैकघनत्वेनापि तत्सन्तताध्ययनाय वचोविषय एव द्विधाकृतम्

Tayorubhayoḥ svarūpaṁ svātantryānandātmakaikaghanatvenāpi tatsantatādhyayanāya vacoviṣaya eva dvidhākṛtam

Even though (api) the essential nature (sva-rūpam) of Them (tayoḥ) both (ubhayoḥ) (is) one compact mass (eka-ghanatvena) composed of (ātmaka) Absolute Freedom (svātantrya)(and) Bliss (ānanda), it is divided into two (dvidhā-kṛtam) —only (eva) in the sphere (viṣaye) of words (vacas)— for its close study (tad-santata-adhyayanāya)

The Self is Absolute Freedom and His Power is Bliss. Both form a compact mass (i.e. omnipresent). In other words, the Highest Reality is always “One without a second”, but, in the world of words It is divided into two for studying It in detail. When this division occurs, the act of coming to recognize or realize the Highest Reality is made easier. So, the very Highest Reality generates this division in the sphere of words as a help for the spiritual aspirants to realize It faster.

आत्मा प्रकाशात्मकशुद्धबोधोऽपि सोऽहमिति वचोविषये स्मृतः

Ātmā prakāśātmakaśuddhabodho’pi so’hamiti vacoviṣaye smṛtaḥ

Although (api) the Self (ātmā) (is) pure (śuddha) Consciousness (bodhaḥ) consisting of (ātmaka) Prakāśa or Light (prakāśa), He (saḥ) is called (smṛtaḥ) “I” (aham iti) in the sphere (viṣaye) of words (vacas)

The Self is pure Consciousness, viz. the Supreme Subject who is never an object. Therefore, He cannot be perceived in the form of “this” or “that”. He cannot even be delineated in thought by any means. Anyway, in the world of words, He is called “I” or also “real I” for the sake of showing that He is higher than the false “I” or ego.

Tantric Reality

Tantra Yoga Kosas - AM 02

आत्मा त्वं गिरिजा मतिः सहचराः प्राणाः शरीरं गृहं पूजा ते विषयोपभोगरचना निद्रा समाधिस्थितिः।
सञ्चारः पदयोः प्रदक्षिणविधिः स्तोत्राणि सर्वा गिरो यद्यत्कर्म करोमि तत्तदखिलं शम्भो तवाराधनम्॥

Ātmā tvaṃ Girijā matiḥ sahacarāḥ prāṇāḥ śarīraṃ gṛham
Pūjā te viṣayopabhoga-racanā nidrā samādhi-sthitiḥ |
Sañcāraḥ padayoḥ pradakṣiṇa-vidhiḥ stotrāṇi sarvā giraḥ
Yad-yat karma karomi tat-tad-akhilaṁ Śambho tavārādhanam ||

You (tvam) (are) the Self (ātmā) and Girijā –an epithet of Pārvatī, Śiva’s wife, meaning “mountain-born”– (girijā) (is) the intelligence (matiḥ). The vital energies (prāṇāḥ) (are Your)companions (sahacarāḥ). The body (śarīram) (is Your) house (gṛham). Worship (pūjā) of You (te) is prepared (racanā) with the objects (viṣaya) (known as sensual) enjoyments (upabhoga). Sleep (nidrā) (is Your) state (sthitiḥ) of Samādhi –i.e. perfect concentration or absorption– (samādhi). (My) wandering (sañcāraḥ) (is) the ceremony (vidhiḥ) of circumambulation from left to right (pradakṣiṇa) of (Your) feet (padayoḥ) –this act is generally done as a token of respect–. All (sarvāḥ) (my) words (giraḥ) (are) hymns of praise (of You) (stotrāṇi). Whatever (yad yad) action (karma) I do (karomi), all (akhilam) that (tad tad) is adoration (ārādhanam) of You (tava), oh Śambhu — an epithet of Śiva meaning “beneficent, benevolent”.

This Self is an embodiment of the Light of Consciousness; it is Śiva, free and autonomous. As an independent play of intense joy, the Divine conceals its own true nature [by manifesting plurality], and may also choose to reveal its fullness once again at any time. All that exists, throughout all time and beyond, is one infinite divine Consciousness, free and blissful, which projects within the field of its awareness a vast multiplicity of apparently differentiated subjects and objects: each object an actualization of a timeless potentiality inherent in the Light of Consciousness, and each subject the same plus a contracted locus of self-awareness. This creation, a divine play, is the result of the natural impulse within Consciousness to express the totality of its self-knowledge in action, an impulse arising from love. The unbounded Light of Consciousness contracts into finite embodied loci of awareness out of its own free will. When those finite subjects then identify with the limited and circumscribed cognitions and circumstances that make up this phase of their existence, instead of identifying with the transindividual overarching pulsation of pure Awareness that is their true nature, they experience what they call “suffering.” To rectify this, some feel an inner urge to take up the path of spiritual gnosis and yogic practice, the purpose of which is to undermine their misidentification and directly reveal within the immediacy of awareness the fact that the divine powers of Consciousness, Bliss, Willing, Knowing, and Acting comprise the totality of individual experience as well – thereby triggering a recognition that one’s real identity is that of the highest Divinity, the Whole in every part. This experiential gnosis is repeated and reinforced through various means until it becomes the nonconceptual ground of every moment of experience, and one’s contracted sense of self and separation from the Whole is finally annihilated in the incandescent radiance of the complete expansion into perfect wholeness. Then one’s perception fully encompasses the reality of a universe dancing ecstatically in the animation of its completely perfect divinity.”

 

Belief Networks “Acyclicity”. Thought of the Day 69.0

Belief networks are used to model uncertainty in a domain. The term “belief networks” encompasses a whole range of different but related techniques which deal with reasoning under uncertainty. Both quantitative (mainly using Bayesian probabilistic methods) and qualitative techniques are used. Influence diagrams are an extension to belief networks; they are used when working with decision making. Belief networks are used to develop knowledge based applications in domains which are characterised by inherent uncertainty. Increasingly, belief network techniques are being employed to deliver advanced knowledge based systems to solve real world problems. Belief networks are particularly useful for diagnostic applications and have been used in many deployed systems. The free-text help facility in the Microsoft Office product employs Bayesian belief network technology. Within a belief network the belief of each node (the node’s conditional probability) is calculated based on observed evidence. Various methods have been developed for evaluating node beliefs and for performing probabilistic inference. Influence diagrams, which are an extension of belief networks, provide facilities for structuring the goals of the diagnosis and for ascertaining the value (the influence) that given information will have when determining a diagnosis. In influence diagrams, there are three types of node: chance nodes, which correspond to the nodes in Bayesian belief networks; utility nodes, which represent the utilities of decisions; and decision nodes, which represent decisions which can be taken to influence the state of the world. Influence diagrams are useful in real world applications where there is often a cost, both in terms of time and money, in obtaining information.

The basic idea in belief networks is that the problem domain is modelled as a set of nodes interconnected with arcs to form a directed acyclic graph. Each node represents a random variable, or uncertain quantity, which can take two or more possible values. The arcs signify the existence of direct influences between the linked variables, and the strength of each influence is quantified by a forward conditional probability.

The Belief Network, which is also called the Bayesian Network, is a directed acyclic graph for probabilistic reasoning. It defines the conditional dependencies of the model by associating each node X with a conditional probability P(X|Pa(X)), where Pa(X) denotes the parents of X. Here are two of its conditional independence properties:

1. Each node is conditionally independent of its non-descendants given its parents.

2. Each node is conditionally independent of all other nodes given its Markov blanket, which consists of its parents, children, and children’s parents.

The inference of Belief Network is to compute the posterior probability distribution

P(H|V) = P(H,V)/ ∑HP(H,V)

where H is the set of the query variables, and V is the set of the evidence variables. Approximate inference involves sampling to compute posteriors. The Sigmoid Belief Network is a type of the Belief Network such that

P(Xi = 1|Pa(Xi)) = σ( ∑Xj ∈ Pa(Xi) WjiXj + bi)

where Wji is the weight assigned to the edge from Xj to Xi, and σ is the sigmoid function.

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Accelerated Capital as an Anathema to the Principles of Communicative Action. A Note Quote on the Reciprocity of Capital and Ethicality of Financial Economics

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Markowitz portfolio theory explicitly observes that portfolio managers are not (expected) utility maximisers, as they diversify, and offers the hypothesis that a desire for reward is tempered by a fear of uncertainty. This model concludes that all investors should hold the same portfolio, their individual risk-reward objectives are satisfied by the weighting of this ‘index portfolio’ in comparison to riskless cash in the bank, a point on the capital market line. The slope of the Capital Market Line is the market price of risk, which is an important parameter in arbitrage arguments.

Merton had initially attempted to provide an alternative to Markowitz based on utility maximisation employing stochastic calculus. He was only able to resolve the problem by employing the hedging arguments of Black and Scholes, and in doing so built a model that was based on the absence of arbitrage, free of turpe-lucrum. That the prescriptive statement “it should not be possible to make sure profits”, is a statement explicit in the Efficient Markets Hypothesis and in employing an Arrow security in the context of the Law of One Price. Based on these observations, we conject that the whole paradigm for financial economics is built on the principle of balanced reciprocity. In order to explore this conjecture we shall examine the relationship between commerce and themes in Pragmatic philosophy. Specifically, we highlight Robert Brandom’s (Making It Explicit Reasoning, Representing, and Discursive Commitment) position that there is a pragmatist conception of norms – a notion of primitive correctnesses of performance implicit in practice that precludes and are presupposed by their explicit formulation in rules and principles.

The ‘primitive correctnesses’ of commercial practices was recognised by Aristotle when he investigated the nature of Justice in the context of commerce and then by Olivi when he looked favourably on merchants. It is exhibited in the doux-commerce thesis, compare Fourcade and Healey’s contemporary description of the thesis Commerce teaches ethics mainly through its communicative dimension, that is, by promoting conversations among equals and exchange between strangers, with Putnam’s description of Habermas’ communicative action based on the norm of sincerity, the norm of truth-telling, and the norm of asserting only what is rationally warranted …[and] is contrasted with manipulation (Hilary Putnam The Collapse of the Fact Value Dichotomy and Other Essays)

There are practices (that should be) implicit in commerce that make it an exemplar of communicative action. A further expression of markets as centres of communication is manifested in the Asian description of a market brings to mind Donald Davidson’s (Subjective, Intersubjective, Objective) argument that knowledge is not the product of a bipartite conversations but a tripartite relationship between two speakers and their shared environment. Replacing the negotiation between market agents with an algorithm that delivers a theoretical price replaces ‘knowledge’, generated through communication, with dogma. The problem with the performativity that Donald MacKenzie (An Engine, Not a Camera_ How Financial Models Shape Markets) is concerned with is one of monism. In employing pricing algorithms, the markets cannot perform to something that comes close to ‘true belief’, which can only be identified through communication between sapient humans. This is an almost trivial observation to (successful) market participants, but difficult to appreciate by spectators who seek to attain ‘objective’ knowledge of markets from a distance. To appreciate the relevance to financial crises of the position that ‘true belief’ is about establishing coherence through myriad triangulations centred on an asset rather than relying on a theoretical model.

Shifting gears now, unless the martingale measure is a by-product of a hedging approach, the price given by such martingale measures is not related to the cost of a hedging strategy therefore the meaning of such ‘prices’ is not clear. If the hedging argument cannot be employed, as in the markets studied by Cont and Tankov (Financial Modelling with Jump Processes), there is no conceptual framework supporting the prices obtained from the Fundamental Theorem of Asset Pricing. This lack of meaning can be interpreted as a consequence of the strict fact/value dichotomy in contemporary mathematics that came with the eclipse of Poincaré’s Intuitionism by Hilbert’s Formalism and Bourbaki’s Rationalism. The practical problem of supporting the social norms of market exchange has been replaced by a theoretical problem of developing formal models of markets. These models then legitimate the actions of agents in the market without having to make reference to explicitly normative values.

The Efficient Market Hypothesis is based on the axiom that the market price is determined by the balance between supply and demand, and so an increase in trading facilitates the convergence to equilibrium. If this axiom is replaced by the axiom of reciprocity, the justification for speculative activity in support of efficient markets disappears. In fact, the axiom of reciprocity would de-legitimise ‘true’ arbitrage opportunities, as being unfair. This would not necessarily make the activities of actual market arbitrageurs illicit, since there are rarely strategies that are without the risk of a loss, however, it would place more emphasis on the risks of speculation and inhibit the hubris that has been associated with the prelude to the recent Crisis. These points raise the question of the legitimacy of speculation in the markets. In an attempt to understand this issue Gabrielle and Reuven Brenner identify the three types of market participant. ‘Investors’ are preoccupied with future scarcity and so defer income. Because uncertainty exposes the investor to the risk of loss, investors wish to minimise uncertainty at the cost of potential profits, this is the basis of classical investment theory. ‘Gamblers’ will bet on an outcome taking odds that have been agreed on by society, such as with a sporting bet or in a casino, and relates to de Moivre’s and Montmort’s ‘taming of chance’. ‘Speculators’ bet on a mis-calculation of the odds quoted by society and the reason why speculators are regarded as socially questionable is that they have opinions that are explicitly at odds with the consensus: they are practitioners who rebel against a theoretical ‘Truth’. This is captured in Arjun Appadurai’s argument that the leading agents in modern finance believe in their capacity to channel the workings of chance to win in the games dominated by cultures of control . . . [they] are not those who wish to “tame chance” but those who wish to use chance to animate the otherwise deterministic play of risk [quantifiable uncertainty]”.

In the context of Pragmatism, financial speculators embody pluralism, a concept essential to Pragmatic thinking and an antidote to the problem of radical uncertainty. Appadurai was motivated to study finance by Marcel Mauss’ essay Le Don (The Gift), exploring the moral force behind reciprocity in primitive and archaic societies and goes on to say that the contemporary financial speculator is “betting on the obligation of return”, and this is the fundamental axiom of contemporary finance. David Graeber (Debt The First 5,000 Years) also recognises the fundamental position reciprocity has in finance, but where as Appadurai recognises the importance of reciprocity in the presence of uncertainty, Graeber essentially ignores uncertainty in his analysis that ends with the conclusion that “we don’t ‘all’ have to pay our debts”. In advocating that reciprocity need not be honoured, Graeber is not just challenging contemporary capitalism but also the foundations of the civitas, based on equality and reciprocity. The origins of Graeber’s argument are in the first half of the nineteenth century. In 1836 John Stuart Mill defined political economy as being concerned with [man] solely as a being who desires to possess wealth, and who is capable of judging of the comparative efficacy of means for obtaining that end.

In Principles of Political Economy With Some of Their Applications to Social Philosophy, Mill defended Thomas Malthus’ An Essay on the Principle of Population, which focused on scarcity. Mill was writing at a time when Europe was struck by the Cholera pandemic of 1829–1851 and the famines of 1845–1851 and while Lord Tennyson was describing nature as “red in tooth and claw”. At this time, society’s fear of uncertainty seems to have been replaced by a fear of scarcity, and these standards of objectivity dominated economic thought through the twentieth century. Almost a hundred years after Mill, Lionel Robbins defined economics as “the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses”. Dichotomies emerge in the aftermath of the Cartesian revolution that aims to remove doubt from philosophy. Theory and practice, subject and object, facts and values, means and ends are all separated. In this environment ex cathedra norms, in particular utility (profit) maximisation, encroach on commercial practice.

In order to set boundaries on commercial behaviour motivated by profit maximisation, particularly when market uncertainty returned after the Nixon shock of 1971, society imposes regulations on practice. As a consequence, two competing ethics, functional Consequential ethics guiding market practices and regulatory Deontological ethics attempting stabilise the system, vie for supremacy. It is in this debilitating competition between two essentially theoretical ethical frameworks that we offer an explanation for the Financial Crisis of 2007-2009: profit maximisation, not speculation, is destabilising in the presence of radical uncertainty and regulation cannot keep up with motivated profit maximisers who can justify their actions through abstract mathematical models that bare little resemblance to actual markets. An implication of reorienting financial economics to focus on the markets as centres of ‘communicative action’ is that markets could become self-regulating, in the same way that the legal or medical spheres are self-regulated through professions. This is not a ‘libertarian’ argument based on freeing the Consequential ethic from a Deontological brake. Rather it argues that being a market participant entails restricting norms on the agent such as sincerity and truth telling that support knowledge creation, of asset prices, within a broader objective of social cohesion. This immediately calls into question the legitimacy of algorithmic/high- frequency trading that seems an anathema in regard to the principles of communicative action.

Vedic Mathematics: Sixteen Sutras and their Corollaries

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The divergence embraces everything other than the fact of intuition itself – the object and field of intuitive vision, the method of working out experience and rendering it to the intellect. The modern method is to get the intuition by suggestion from an appearance in life or nature or from a mental idea and even if the source of the intuition ie the soul, the method at once relates it to a support external to the soul. The ancient Indian method of knowledge had for its business to disclose something of the Self, the Infinite or the Divine to the regard of the soul – the Self through its expressions, the infinite through its finite symbols and the Divine through his powers. The process was one of Integral knowledge and in its subordinate ranges was instrumental in revealing the truths of cosmic phenomena and these truths mere utilised for worldly ends.

Tirthaji_S.B.K. Vedic mathematics or sixteen simple mathematical formulae from the Vedas

Rants of the Undead God: Instrumentalism. Thought of the Day 68.1

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Hilbert’s program has often been interpreted as an instrumentalist account of mathematics. This reading relies on the distinction Hilbert makes between the finitary part of mathematics and the non-finitary rest which is in need of grounding (via finitary meta-mathematics). The finitary part Hilbert calls “contentual,” i.e., its propositions and proofs have content. The infinitary part, on the other hand, is “not meaningful from a finitary point of view.” This distinction corresponds to a distinction between formulas of the axiomatic systems of mathematics for which consistency proofs are being sought. Some of the formulas correspond to contentual, finitary propositions: they are the “real” formulas. The rest are called “ideal.” They are added to the real part of our mathematical theories in order to preserve classical inferences such as the principle of the excluded middle for infinite totalities, i.e., the principle that either all numbers have a given property or there is a number which does not have it.

It is the extension of the real part of the theory by the ideal, infinitary part that is in need of justification by a consistency proof – for there is a condition, a single but absolutely necessary one, to which the use of the method of ideal elements is subject, and that is the proof of consistency; for, extension by the addition of ideals is legitimate only if no contradiction is thereby brought about in the old, narrower domain, that is, if the relations that result for the old objects whenever the ideal objects are eliminated are valid in the old domain. Weyl described Hilbert’s project as replacing meaningful mathematics by a meaningless game of formulas. He noted that Hilbert wanted to “secure not truth, but the consistency of analysis” and suggested a criticism that echoes an earlier one by Frege – why should we take consistency of a formal system of mathematics as a reason to believe in the truth of the pre-formal mathematics it codifies? Is Hilbert’s meaningless inventory of formulas not just “the bloodless ghost of analysis? Weyl suggested that if mathematics is to remain a serious cultural concern, then some sense must be attached to Hilbert’s game of formulae. In theoretical physics we have before us the great example of a [kind of] knowledge of completely different character than the common or phenomenal knowledge that expresses purely what is given in intuition. While in this case every judgment has its own sense that is completely realizable within intuition, this is by no means the case for the statements of theoretical physics. Hilbert suggested that consistency is not the only virtue ideal mathematics has –  transfinite inference simplifies and abbreviates proofs, brevity and economy of thought are the raison d’être of existence proofs.

Hilbert’s treatment of philosophical questions is not meant as a kind of instrumentalist agnosticism about existence and truth and so forth. On the contrary, it is meant to provide a non-skeptical and positive solution to such problems, a solution couched in cognitively accessible terms. And, it appears, the same solution holds for both mathematical and physical theories. Once new concepts or “ideal elements” or new theoretical terms have been accepted, then they exist in the sense in which any theoretical entities exist. When Weyl eventually turned away from intuitionism, he emphasized the purpose of Hilbert’s proof theory, not to turn mathematics into a meaningless game of symbols, but to turn it into a theoretical science which codifies scientific (mathematical) practice. The reading of Hilbert as an instrumentalist goes hand in hand with a reading of the proof-theoretic program as a reductionist project. The instrumentalist reading interprets ideal mathematics as a meaningless formalism, which simplifies and “rounds out” mathematical reasoning. But a consistency proof of ideal mathematics by itself does not explain what ideal mathematics is an instrument for.

On this picture, classical mathematics is to be formalized in a system which includes formalizations of all the directly verifiable (by calculation) propositions of contentual finite number theory. The consistency proof should show that all real propositions which can be proved by ideal methods are true, i.e., can be directly verified by finite calculation. Actual proofs such as the ε-substitution procedure are of such a kind: they provide finitary procedures which eliminate transfinite elements from proofs of real statements. In particular, they turn putative ideal derivations of 0 = 1 into derivations in the real part of the theory; the impossibility of such a derivation establishes consistency of the theory. Indeed, Hilbert saw that something stronger is true: not only does a consistency proof establish truth of real formulas provable by ideal methods, but it yields finitary proofs of finitary general propositions if the corresponding free-variable formula is derivable by ideal methods.

Epistemological Constraints to Finitism. Thought of the Day 68.0

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Hilbert’s substantial philosophical claims about the finitary standpoint are difficult to flesh out. For instance, Hilbert appeals to the role of Kantian intuition for our apprehension of finitary objects (they are given in the faculty of representation). Supposing one accepts this line of epistemic justification in principle, it is plausible that the simplest examples of finitary objects and propositions, and perhaps even simple cases of finitary operations such as concatenations of numerals can be given a satisfactory account.

Of crucial importance to both an understanding of finitism and of Hilbert’s proof theory is the question of what operations and what principles of proof should be allowed from the finitist standpoint. That a general answer is necessary is clear from the demands of Hilbert’s proof theory, i.e., it is not to be expected that given a formal system of mathematics (or even a single sequence of formulas) one can “see” that it is consistent (or that it cannot be a genuine derivation of an inconsistency) the way we can see, e.g., that || + ||| = ||| + ||. What is required for a consistency proof is an operation which, given a formal derivation, transforms such a derivation into one of a special form, plus proofs that the operation in fact succeeds in every case and that proofs of the special kind cannot be proofs of an inconsistency.

Hilbert said that intuitive thought “includes recursion and intuitive induction for finite existing totalities.” All of this in its application in the domain of numbers, can be formalized in a system known as primitive recursive arithmetic (PRA), which allows definitions of functions by primitive recursion and induction on quantifier-free formulas. However, Hilbert never claimed that only primitive recursive operations count as finitary. Although Hilbert and his collaborators used methods which go beyond the primitive recursive and accepted them as finitary, it is still unclear whether they (a) realized that these methods were not primitive recursive and (b) whether they would still have accepted them as finitary if they had. The conceptual issue is which operations should be considered as finitary. Since Hilbert was less than completely clear on what the finitary standpoint consists in, there is some leeway in setting up the constraints, epistemological and otherwise, an analysis of finitist operation and proof must fulfill. Hilbert characterized the objects of finitary number theory as “intuitively given,” as “surveyable in all their parts,” and said that their having basic properties must “exist intuitively” for us. This characterization of finitism as primarily to do with intuition and intuitive knowledge has been emphasized in that what can count as finitary on this understanding is not more than those arithmetical operations that can be defined from addition and multiplication using bounded recursion.

Rejecting the aspect of representability in intuition as the hallmark of the finitary; one could take finitary reasoning to be “a minimal kind of reasoning supposed by all non-trivial mathematical reasoning about numbers” and analyze finitary operations and methods of proof as those that are implicit in the very notion of number as the form of a finite sequence. This analysis of finitism is supported by Hilbert’s contention that finitary reasoning is a precondition for logical and mathematical, indeed, any scientific thinking.