§2.3 — The blockchain is not ‘mere’ code – even highly automated code. It cannot be anything, determinable within an ontology established at a superior level to itself. Nakamoto Consensus is less an *object* for philosophy than a virtual *criterion*: a fundamental, obliquely mechanized decision procedure for settling the nature of truth. In other words, Bitcoin is a transcendental operation, before becoming the topic for one. The primary meaning of ‘transcendental’ is *ultimate*, which can be clarified negatively by *the absence of any higher or superior tribunal*. There is no place from which to consistently or authoritatively *second-guess* the blockchain. By implementing a “fully peer-to-peer” system, which subtracts the role of “third party” monitoring and adjudication, the Bitcoin protocol automatically places itself beyond external oversight. Its criterion of validation is radically *immanent*. The task of this work, therefore, is not to subject Bitcoin to philosophical judgment, but rather to elaborate the lessons of Bitcoin through a philosophical *hash*.

§2.31 — *Hashing* is the coding process of most unmistakable relevance to the docking of Bitcoin onto the language of philosophy. Hashing is not only – though it is overwhelmingly – what running Bitcoin involves. It is also, in addition, an automatic translation procedure, and a categorical scheme implemented in software. Hash-functions are codes, and thus mappings (from ‘keys’ to ‘values’), or systematic text conversions. Hashing an initial input text produces a compressed translation (the ‘hash’, ‘digest’, or ‘tag’). As with any process of filing, the value of the hash depends upon constriction. A comparative plethora of initial elements is reduced to a smaller range of terminal variation. Any hash is inseparable, therefore, from an *economization*. Because a hash sorts inputs into output ‘buckets’ it is already, and intrinsically, also a *categorization*. Finally, any hash is inevitably a kind of *cipher*. It converts an input text into other terms. The existence of specifically cryptographic hashes is, then, to be expected.

§2.311 — The cryptographic hash function adopted as a basic building block by the Bitcoin Protocol is the 256-bit (32-byte) Secure Hash Algorithm neatly abbreviated as SHA-256. It belongs to the SHA 2 family of such algorithms, designed by the NSA, and first published in 2001. Within the Bitcoin system, SHA-256 sets the proof-of-work test that secures the currency through the same process in which it is systematically hacked.* The cryptographic challenge is designed to be (arduously) *puzzled out*, automatically modified, and re-posed. Each such event is a basic unit of time, or elementary episode, determining a block on the chain. *Hashing* and *mining* are made synonymous, as Bitcoin’s primary process. The hashing cycle establishes an ultimate, unsurpassable, transcendental, or *chronogenic* function.

§2.312 — A cultural side-product of the Bitcoin protocol, then, is a cryptographic definition of time. Punctual-geometric ‘now’, as marked on a ‘time-line’, is replaced by an atomic *unit of irreducible duration*, coinciding with the completion of a block, and ordered successively on the chain. Between duration and succession, the relation is *synthetic*. The blockchain is constituted by a series of durations, which are not inter-convertible, or mathematically transformable into each other. Hash-time has ceased to be accurately representable as a dimension. A time-line merely analogizes it, to what is an ultimately inadequate level of definite fidelity.

§2.313 — The weakly-formalized hash function employed in this book is Kantian critique. It latches upon input text extracted from the cultural agitation attending crypto-currency techonomics, and outputs a digest in the (partially submerged) mainstream language of philosophy. Peer-to-peer flatness is hashed into *immanence*, ‘trusted third parties’ into metaphysical constructs of *transcendence*. Since the mid-19th century, the primary impetus of transcendental philosophy has been directed to the materialization of critique. Academic philosophy, almost by definition, has not registered this trend accurately. It has been through the advances and errors of cybernetics and historical materialism that critical modernity has been charted. The dominant academic traditions of linguistic philosophy (in the Anglophone world) and phenomenology (in Continental Europe) have only weakly reflected such developments. When resistance to materialization is a guild imperative, even the most sincere attempts to *bring thought into compliance with the real process* founder, through institutional necessity. There is not, in any case, solid ground upon which to idealize such sincerity unduly, since its orientation is essentially misconceived. Transcendence poses real problems – *obstacles* – requiring techonomic solutions, rather than mere conceptual exorcism. Immanentization is the product of a diagonal process, leading *through* the exteriority of the machine. ‘Armchair philosophy’ should not, therefore, be opposed to an armchair skepticism, but to the history of cryptography, in its broadest possible conception, which relates the hidden and unhidden to the irreversible emergence of real capabilities.

§2.32 — The ultimate foundation of the Kantian critical philosophy is a *difference*, drawn between objects and their conditions of possibility. Items of competent attention are framed in a way that cannot itself be successfully itemized. The display frame cannot be displayed. Confusion between (*empirical*) objects and their (*transcendental*) conditions of possibility, most typically exemplified by the attempt to apprehend the latter as if they were the former, is taken to define *speculative metaphysics* (or pure theoretical reason) – which is conceived, rigorously, as a persistent yet futile misapplication of intelligence to pseudo-problems essentially exceeding its capabilities. The rest is detail.

§2.321 — To objectify the transcendental bases of objectivity, for instance, in the posing of a metaphysical question about the ‘nature’ of space, time, or causality, is to lead thought into hopeless error, whose symptoms are irresolvable dilemmas (contradictions, or *antinomies*). The systematic enumeration of these cognitive dead-ends is the task of *transcendental dialectic*. It was Kant’s contention that such Quixotic questions – addressed to the conditions of objectivity as if they were themselves objects – had dominated and fatally distracted philosophy up to his own time. The repudiation of such error is, at its most elementary – and considered here, initially, solely in its theoretical employment – the primary product of Kantian *critique*.

§2.322 — Critique sets limits. It also *eliminates*. That is why the critique of metaphysics has been found to be isomorphic with a socio-political project of subtraction, with an inclination towards anarchism. The promotion by Satoshi Nakamoto of a platform for peer-to-peer transactions independent of all oversight by “trusted third parties” is the continuation of critique into electronic networks. The same impulse is more widely recognized as ‘disintermediation’. It complies with the quintessentially modernistic project of *immanentization*. Transcendent ‘grounds’ of authority are identified, delimited, routed-around, obsolesced, and finally extirpated. Modernity, as the work of critique, produces *formal flatness*.

§2.323 — Considered as a positive philosophical discovery, the transcendental coincides with the *synthetic a priori*. Like all great things in the domain of thought, this hybrid concept is quasi-paradoxical. It denotes a field of *non-factual discovery* – a genetic particularity of the universal – at once necessary but non-obvious, epitomized by the mathematical theorem. Synthetic *a priori* truths are secular revelations. Contingent in their acquisition, but then necessary in their preservation, they constitute the sole positive ratchet in the accumulation of knowledge, describing an asymmetry – or ‘arrow’ – proper to epistemology: a one-way, or unilateral, fatality. Such discoveries are arduously amassed, but then invulnerable to dissipation. They are in this way indispensable to the comprehension of historical time, and can be considered as *products of unlimited application*. The blockchain is exemplary. A *cryptic*, or radically non-obvious solution to a problem we will later explore attentively, it is – *subsequent to its formalization* – culturally indispensable. It ‘cannot be un-invented’. This is true to such an extent that it appears as an eternal mathematical fact, wholly impervious to the ravages of empirical fortuity. To de-realize the blockchain would be to unmake the universe (or at least, to collapse what is – transcendentally or inescapably – *for us* the universe). What is done transcendentally cannot be undone, without radical time-violation. The crypto-current permits no repudiation. The units of synthetic *a priori* knowledge production are *laws*, in the very strongest defensible sense of this term, in which their descent *from*, and simultaneous irreducibility *to*, any particular *cases* is insisted upon. This ratchet-structure makes the synthetic *a priori* – or some adequate analog – indispensable to any rigorous conceptual decompression of the notion of time.

§2.33 — Formulation of the synthetic *a priori* exemplifies the philosophical deployment of diagonal argument. It crosses through the previously uncontroversial, and implicitly exhaustive, distinction between the analytic *a priori* and the synthetic *a posteriori* at a slant. By first decompressing this binary structure into a (two-dimensional) table, or matrix, and then registering a hybrid term that could not otherwise be identified, such diagonal argument approximates to a mechanized conceptual production.** Its iron necessity, however, is strictly retroactive. Were it susceptible to confident anticipation, in accordance with a formula, it would reduce to an analytic statement.*** Within this process, essentially, the distinction between discovery and innovation is itself diagonalized.****

§2.34 — It is from irreversibility – of the one-way (or ‘trap-door’) crypto-function, the thermodynamic gradient, and ultimately of absolute time – that the reliable principle of analytic-synthetic distinction can be isolated. A mathematical proof is easier to confirm than to construct. Prime numbers are easily multiplied, but their product is time-consuming to factorize. Bitcoin blocks are easy to check, but hard to mine. In each case there is a distinction between analytical facility and (comparative) synthetic intractability. When cryptographically re-conceived, analysis and synthesis co-produce a ratchet. Adam Back (on Twitter) describes the mechanized contractual commitment that exploits this gradient as “computational irrevocability”. Like a carnivorous plant, it is easy to enter, but then difficult to escape. History is a Venus flytrap, self-abstracted beyond botany.

§2.35 — The essential and continuous features of critique, abstractly apprehended, therefore, reduce to (1) the articulation of transcendental-empirical difference, (2) subtraction of the transcendent object in the name of immanence, and (3) temporalization of philosophical problematics (onto the ultimate gradient, or asymmetric distance, of absolute succession). In combination, these elements draw an abstract diagonal line, or diagram of time, which Kant called a *schema*. The schema practically describes, or protracts, the irreducible *difference* between the transcendental and empirical as a process of conceptual production without transcendent dependency.

* Bitcoin establishes security by continuously attacking itself. Each intrinsic ‘tick’ of Bitcoin time corresponds to an *unlocking*. It is only through being recurrently *broken into*, that the system secures itself. The conceptual radicality of this cryptographic innovation remains under-appreciated. It is further examined in Chapter Six.

** Due – in part – to the residual obscurity of philosophy’s critical enterprise, even to itself, the model of diagonal argument remains Cantor’s demonstration of the uncountable infinities. The formalization of this argument dates back to 1874, with the publication of Cantor’s paper ‘*Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen*’ (‘On a Property of the Collection of All Real Algebraic Numbers’). Every Real number – including the integers – can be described in a form with infinite decimal expansion (since 1/3 x 3 = 1). This notational principle enables each entry, even in a complete enumeration of an infinite numerical series, to be mapped onto a decimal position in any number. Thus the *n*th number can be plotted onto the *n*th *place* of each number (irrespective of modulus), enabling the diagonal construction that carries Cantor’s demonstration, whereby a number that cannot yet have been enumerated is discovered / invented, differing from the *n*th number in its *n*th place. This diagonal operation can, of course, be infinitely reiterated. It confirms the incompleteness of any numerical series, however dense, and thus rigorously establishes the existence of uncountable infinity. Gödelian incompleteness, and Church-Turing non-computability, each reconstitute the same abstract diagonal. The Cantorian diagonalization matrix, since it processes infinite series (in two dimensions), is necessarily notional (virtual). It cannot be actually constructed. It establishes an abstract procedure, absolutely intractable to complete mechanical execution. Compact philosophical diagonalizations, of the Kantian type, nurse comparable infinities, more obscurely. Such incompletely-actualizable ‘outcomes’ betray the encounter with an abstract (limit) trapdoor-function, or fundamental asymmetry, which partially expresses an absolute excess. Diagonalization, in the methodical production of an indicative conceptual surplus, inevitably stumbles upon *time-in-itself* while also – and no less inevitability – demonstrating the limits of its staging. The discovery (synthesis) is precisely: *It cannot be that there is nothing left to do*. This apparently simple formula has proven harder to process than might initially seem imaginable.

*** Echoes of the Kantian distinction between the *a priori* and *a posteriori* are evident in the Bayesian vocabulary of ‘priors’ and ‘posteriors’. This resonance is by no means devoid of philosophical substance. That Bayesianism is the critical (or ‘Copernican’) revolution proper to the theory of probability is marked by its programmatic attachment to the *subjectivety* of probabilistic inference, in a recognizably Kantian sense. Probabilities are not objective characteristics (frequencies), but estimations. Yet even if this structure of probability *as such* is properly transcendental, particular Bayesian priors are empirical (as indicated most obviously by their revisability), and thus do not constitute architectonic elements for conditions of possibility in general.

**** Peter Galison captures the diagonal inclination well: “Raw empiricism was avoided as woefully inadequate to account for the generality and extent of scientific knowledge. Pure idealism (reducing reality to mental life) could not explain the concordance of ideas with the world. Drawing strongly on the Kant revival underway in Germany, [Emile] Boutroux and his circle rejected both the extremes of idealism and empiricism. Taking science and the humanities to be inextricably bound, these philosophers saw both structured by an active role for the mind and a suspicion toward the purely metaphysical. In his encounters with August Calinon’s work on the philosophical foundations of physics, Poincaré walked this philosophical middle line toward the problem of simultaneity.” *Einstein’s Clocks, Poincaré’s Maps: Empires of Time* (2003), p.81.

You have two footnotes labeled with three asterisks.

Buckets are more a non-cryptographic hashing algorithm thing, where the hash value is used to place objects in an area of computer memory for later retrieval. With cryptographic hashing algorithms in practical usage (with the number of items we are likely to hash before the sun explodes), if two items hash to the same “bucket” that’s a hash collision and a security problem.

The Bitcoin whitepaper talks about a “timestamp server”, not a blockchain, so Bitcoin’s cryptographic production of time viewed as part of the *technical* system is more than a side-effect, although in the sense of not being the primary purpose of the system (p2p money) it still is.

Culturally, yes, is has been interesting to watch people intuiting time relative to block production and seeing the effect that difficulty adjustments and statistically expected but intuition-busting variations in block production time have on them.

Cryptographic hashes produce names or identities rather than *useful* categories. If the hash is not a category of one something has gone wrong.

Thanks Rob. Will digest these remarks carefully down the line.

last note is *** too, probably should be ****

“thus rigorously establishes the existence of uncountable infinity”

“It cannot be actually constructed.”

Constructivist (which, IMO, is how math should be done after Kant) interpretation (as e.g. given by Errett Bishop in his _Foundations of Constructive Analysis_, Ch2) of the diagonal argument is that it establishes subcountability.

Bishop’s version is a strict construction. Given a program P that computes a sequence of real numbers, Bishop’s procedure computes a real number r that is not produced by P. Since P is arbitrary, this proves that there’s no program that comprehensively enumerates computable real numbers* even though these are a subset of all programs, which can be completely enumerated by Godel numbering. Hence SUBcountable rather than uncountable.

An amusing aspect of defining real numbers in terms of programs is that it turns out you can even identify uncomputable real numbers in this way: “[A] Specker sequence is a computable, monotonically increasing, bounded sequence of rational numbers whose supremum is not a computable real number.” Of course suprema for all bounded sets exist by fiat in non-constructive math, so they don’t particularly care if digits of a number, that supposedly exists, cannot be computed.

* Where a real number x is conceived as a program p(n) that computes a definite nth rational approximation given positive integer n as input: |x – p(n)| ≤ 1/n. In this construction, equality operation (comparison) is given by the criterion: p_1 = p_2 if |p_1(n) – p_2(n)| ≤ 2/n, n in Z+