Hypostatic Abstraction. Thought of the Day 138.0


Hypostatic abstraction is linguistically defined as the process of making a noun out of an adjective; logically as making a subject out of a predicate. The idea here is that in order to investigate a predicate – which other predicates it is connected to, which conditions it is subjected to, in short to test its possible consequences using Peirce’s famous pragmatic maxim – it is necessary to posit it as a subject for investigation.

Hypostatic abstraction is supposed to play a crucial role in the reasoning process for several reasons. The first is that by making a thing out of a thought, it facilitates the possibility for thought to reflect critically upon the distinctions with which it operates, to control them, reshape them, combine them. Thought becomes emancipated from the prison of the given, in which abstract properties exist only as Husserlian moments, and even if prescission may isolate those moments and induction may propose regularities between them, the road for thought to the possible establishment of abstract objects and the relations between them seems barred. The object created by a hypostatic abstraction is a thing, but it is of course no actually existing thing, rather it is a scholastic ens rationis, it is a figment of thought. It is a second intention thought about a thought – but this does not, in Peirce’s realism, imply that it is necessarily fictitious. In many cases it may indeed be, but in other cases we may hit upon an abstraction having real existence:

Putting aside precisive abstraction altogether, it is necessary to consider a little what is meant by saying that the product of subjectal abstraction is a creation of thought. (…) That the abstract subject is an ens rationis, or creation of thought does not mean that it is a fiction. The popular ridicule of it is one of the manifestations of that stoical (and Epicurean, but more marked in stoicism) doctrine that existence is the only mode of being which came in shortly before Descartes, in concsequence of the disgust and resentment which progressive minds felt for the Dunces, or Scotists. If one thinks of it, a possibility is a far more important fact than any actuality can be. (…) An abstraction is a creation of thought; but the real fact which is important in this connection is not that actual thinking has caused the predicate to be converted into a subject, but that this is possible. The abstraction, in any important sense, is not an actual thought but a general type to which thought may conform.

The seemingly scepticist pragmatic maxim never ceases to surprise: if we take all possible effects we can conceive an object to have, then our conception of those effects is identical with our conception of that object, the maxim claims – but if we can conceive of abstract properties of the objects to have effects, then they are part of our conception of it, and hence they must possess reality as well. An abstraction is a possible way for an object to behave – and if certain objects do in fact conform to this behavior, then that abstraction is real; it is a ‘real possibility’ or a general object. If not, it may still retain its character of possibility. Peirce’s definitions of hypostatic abstractions now and then confuse this point. When he claims that

An abstraction is a substance whose being consists in the truth of some proposition concerning a more primary substance,

then the abstraction’s existence depends on the truth of some claim concerning a less abstract substance. But if the less abstract substance in question does not exist, and the claim in question consequently will be meaningless or false, then the abstraction will – following that definition – cease to exist. The problem is only that Peirce does not sufficiently clearly distinguish between the really existing substances which abstractive expressions may refer to, on the one hand, and those expressions themselves, on the other. It is the same confusion which may make one shuttle between hypostatic abstraction as a deduction and as an abduction. The first case corresponds to there actually existing a thing with the quality abstracted, and where we consequently may expect the existence of a rational explanation for the quality, and, correlatively, the existence of an abstract substance corresponding to the supposed ens rationis – the second case corresponds to the case – or the phase – where no such rational explanation and corresponding abstract substance has yet been verified. It is of course always possible to make an abstraction symbol, given any predicate – whether that abstraction corresponds to any real possibility is an issue for further investigation to estimate. And Peirce’s scientific realism makes him demand that the connections to actual reality of any abstraction should always be estimated (The Essential Peirce):

every kind of proposition is either meaningless or has a Real Secondness as its object. This is a fact that every reader of philosophy should carefully bear in mind, translating every abstractly expressed proposition into its precise meaning in reference to an individual experience.

This warning is directed, of course, towards empirical abstractions which require the support of particular instances to be pragmatically relevant but could hardly hold for mathematical abstraction. But in any case hypostatic abstraction is necessary for the investigation, be it in pure or empirical scenarios.


Austrian School of Economics: The Praxeological Synthetic. Thought of the Day 135.0


Within the Austrian economics (here, here, here and here), the a priori stance has dominated a tradition running from Carl Menger to Murray Rothbard. The idea here is that the basic structures of economy is entrenched in the more basic structures of human action as such. Nowhere is this more evident than in the work of Ludwig von Mises – his so-called ‘praxeology’, which rests on the fundamental axiom that individual human beings act on the primordial fact that individuals engage in conscious actions toward chosen goals, is built from the idea that all basic laws of economy can be derived apriorically from one premiss: the concept of human action. Of course, this concept is no simple concept, containing within itself purpose, product, time, scarcity of resources, etc. – so it would be more fair to say that economics lies as the implication of the basic schema of human action as such.

Even if the Austrian economists’ conception of the a priori is decidedly objectivist and anti-subjectivist, it is important to remark their insistence on subjectivity within their ontological domain. The Austrian economics tradition is famous exactly for their emphasis on the role of subjectivity in economy. From Carl Menger onwards, they protest against the mainstream economical assumption that the economic agent in the market is fully rational, knows his own preferences in detail, has constant preferences over time, has access to all prices for a given commodity at a given moment, etc. Thus, von Mises’ famous criticism of socialist planned economy is built on this idea: the system of ever-changing prices in the market constitutes a dispersed knowledge about the conditions of resource allocation which is a priori impossible for any single agent – let alone, any central planner’s office – to possess. Thus, their conception of the objective a priori laws of the economic domain perhaps surprisingly had the implication that they warned against a too objectivist conception of economy not taking into account the limits of economic rationality stemming from the general limitations of the capacities of real subjects. Their ensuing liberalism is thus built on a priori conclusions about the relative unpredictability of economics founded on the role played by subjective intentionality. For the same reason, Hayek ended up with a distinction between simple and complex processes, respectively, cutting across all empirical disciplines, where only the former permit precise, predictive, quantitative calculi based on mathemathical modeling while the latter permit only recognition of patterns (which may also be mathematically modeled, to be sure, but without quantitative predictability). It is of paramount importance, though, to distinguish this emphasis on the ineradicable role of subjectivity in certain regional domains from Kantian-like ideas about the foundational role of subjectivity in the construction of knowledge as such. The Austrians are as much subjectivists in the former respect as they are objectivists in the latter. In the history of economics, the Austrians occupy a middle position, being against historicism on the one hand as well as against positivism on the other. Against the former, they insist that a priori structures of economy transgress history which does not possess the power to form institutions at random but only as constrained by a priori structures. And against the latter, they insist that the mere accumulation of empirical data subject to induction will never in itself give rise to the formation of theoretical insights. Structures of intelligible concepts are in all cases necessary for any understanding of empirical regularities – in so far, the Austrian a priori approach is tantamount to a non-skepticist version of the doctrine of ‘theory-ladenness’ of observations.

A late descendant of the Austrian tradition after its emigration to the Anglo-Saxon world (von Mises, Hayek, and Schumpeter were such emigrés) was the anarcho-liberal economist Murray Rothbard, and it is the inspiration from him which allows Barry Smith to articulate the principles underlying the Austrians as ‘fallibilistic apriorism’. Rothbard characterizes in a brief paper what he calls ‘Extreme Apriorism’ as follows:

there are two basic differences between the positivists’ model science of physics on the one hand, and sciences dealing with human actions on the other: the former permits experimental verification of consequences of hypotheses, which the latter do not (or, only to a limited degree, we may add); the former admits of no possibility of testing the premisses of hypotheses (like: what is gravity?), while the latter permits a rational investigation of the premisses of hypotheses (like: what is human action?). This state of affairs makes it possible for economics to derive its basic laws with absolute – a priori – certainty: in addition to the fundamental axiom – the existence of human action – only two empirical postulates are needed: ‘(1) the most fundamental variety of resources, both natural and human. From this follows directly the division of labor, the market, etc.; (2) less important, that leisure is a consumer good’. On this basis, it may e.g. be inferred, ‘that every firm aims always at maximizing its psychic profit’.

Rothbard draws forth this example so as to counterargue traditional economists who will claim that the following proposition could be added as a corollary: ‘that every firm aims always at maximizing its money profit’. This cannot be inferred and is, according to Rothbard, an economical prejudice – the manager may, e.g. prefer for nepotistic reasons to employ his stupid brother even if that decreases the firm’s financial profit possibilities. This is an example of how the Austrians refute the basic premiss of absolute rationality in terms of maximal profit seeking. Given this basis, other immediate implications are:

the means-ends relationship, the time-structure of production, time-preference, the law of diminishing marginal utility, the law of optimum returns, etc.

Rothbard quotes Mises for seeing the fundamental Axiom as a ‘Law of Thought’ – while he himself sees this as a much too Kantian way of expressing it, he prefers instead the simple Aristotelian/Thomist idea of a ‘Law of Reality’. Rothbard furthermore insists that this doctrine is not inherently political – in order to attain the Austrians’ average liberalist political orientation, the preference for certain types of ends must be added to the a priori theory (such as the preference for life over death, abundance over poverty, etc.). This also displays the radicality of the Austrian approach: nothing is assumed about the content of human ends – this is why they will never subscribe to theories about Man as economically rational agent or Man as necessarily economical egotist. All different ends meet and compete on the market – including both desire for profit in one end and idealist, utopian, or altruist goals in the other. The principal interest, in these features of economical theory is the high degree of awareness of the difference between the – extreme – synthetic a priori theory developed, on the one hand, and its incarnation in concrete empirical cases and their limiting conditions on the other.


Metaphysical Continuity in Peirce. Thought of the Day 122.0


Continuity has wide implications in the different parts of Peirce’s architectonics of theories. Time and time again, Peirce refers to his ‘principle of continuity’ which has not immediately anything to do with Poncelet’s famous such principle in geometry, but, is rather, a metaphysical implication taken to follow from fallibilism: if all more or less distinct phenomena swim in a vague sea of continuity then it is no wonder that fallibilism must be accepted. And if the world is basically continuous, we should not expect conceptual borders to be definitive but rather conceive of terminological distinctions as relative to an underlying, monist continuity. In this system, mathematics is first science. Thereafter follows philosophy which is distinguished form purely hypothetical mathematics by having an empirical basis. Philosophy, in turn, has three parts, phenomenology, the normative sciences, and metaphysics. The first investigates solely ‘the Phaneron’ which is all what could be imagined to appear as an object for experience: ‘ by the word phaneron I mean the collective total of all that is in any way or in any sense present to the mind, quite regardless whether it corresponds to any real thing or not.’ (Charles Sanders Peirce – Collected Papers of Charles Sanders Peirce) As is evident, this definition of Peirce’s ‘phenomenology’ is parallel to Husserl’s phenomenological reduction in bracketing the issue of the existence of the phenomenon in question. Even if it thus is built on introspection and general experience, it is – analogous to Husserl and other Brentano disciples at the same time – conceived in a completely antipsychological manner: ‘It religiously abstains from all speculation as to any relations between its categories and physiological facts, cerebral or other.’ and ‘ I abstain from psychology which has nothing to do with ideoscopy.’ (Letter to Lady Welby). The normative sciences fall in three: aesthetics, ethics, logic, in that order (and hence decreasing generality), among which Peirce does not spend very much time on the former two. Aesthetics is the investigation of which possible goals it is possible to aim at (Good, Truth, Beauty, etc.), and ethics how they may be reached. Logic is concerned with the grasping and conservation of Truth and takes up the larger part of Peirce’s interest among the normative sciences. As it deals with how truth can be obtained by means of signs, it is also called semiotics (‘logic is formal semiotics’) which is thus coextensive with theory of science – logic in this broad sense contains all parts of philosophy of science, including contexts of discovery as well as contexts of justification. Semiotics has, in turn, three branches: grammatica speculativa (or stekheiotics), critical logic, and methodeutic (inspired by mediaeval trivium: grammar, logic, and rhetoric). The middle one of these three lies closest to our days’ conception of logic; it is concerned with the formal conditions for truth in symbols – that is, propositions, arguments, their validity and how to calculate them, including Peirce’s many developments of the logic of his time: quantifiers, logic of relations, ab-, de-, and induction, logic notation systems, etc. All of these, however, presuppose the existence of simple signs which are investigated by what is often seen as semiotics proper, the grammatica speculativa; it may also be called formal grammar. It investigates the formal condition for symbols having meaning, and it is here we find Peirce’s definition of signs and his trichotomies of different types of sign aspects. Methodeutic or formal rhetorics, on the other hand, concerns the pragmatical use of the former two branches, that is, the study of how to use logic in a fertile way in research, the formal conditions for the ‘power’ of symbols, that is, their reference to their interpretants; here can be found, e.g., Peirce’s famous definitions of pragmati(ci)sm and his directions for scientific investigation. To phenomenology – again in analogy to Husserl – logic adds the interest in signs and their truth. After logic, metaphysics follows in Peirce’s system, concerning the inventarium of existing objects, conceived in general – and strongly influenced by logic in the Kantian tradition for seeing metaphysics mirroring logic. Also here, Peirce has several proposals for subtypologies, even if none of them seem stable, and under this headline classical metaphysical issues mix freely with generalizations of scientific results and cosmological speculations.

Peirce himself saw this classification in an almost sociological manner, so that the criteria of distinction do not stem directly from the implied objects’ natural kinds, but after which groups of persons study which objects: ‘the only natural lines of demarcation between nearly related sciences are the divisions between the social groups of devotees of those sciences’. Science collects scientists into bundles, because they are defined by their causa finalis, a teleologial intention demanding of them to solve a central problem.

Measured on this definition, one has to say that Peirce himself was not modest, not only does he continuously transgress such boundaries in his production, he frequently does so even within the scope of single papers. There is always, in his writings, a brief distance only from mathematics to metaphysics – or between any other two issues in mathematics and philosophy, and this implies, first, that the investigation of continuity and generality in Peirce’s system is more systematic than any actually existing exposition of these issues in Peirce’s texts, second, that the discussion must constantly rely on cross-references. This has the structural motivation that as soon as you are below the level of mathematics in Peirce’s system, inspired by the Comtean system, the single science receives determinations from three different directions, each science consisting of material and formal aspects alike. First, it receives formal directives ‘from above’, from those more general sciences which stand above it, providing the general frameworks in which it must unfold. Second, it receives material determinations from its own object, requiring it to make certain choices in its use of formal insights from the higher sciences. The cosmological issue of the character of empirical space, for instance, can take from mathematics the different (non-)Euclidean geometries and investigate which of these are fit to describe spatial aspects of our universe, but it does not, in itself, provide the formal tools. Finally, the single sciences receive in practice determinations ‘from below’, from more specific sciences, when their results by means of abstraction, prescission, induction, and other procedures provide insights on its more general, material level. Even if cosmology is, for instance, part of metaphysics, it receives influences from the empirical results of physics (or biology, from where Peirce takes the generalized principle of evolution). The distinction between formal and material is thus level specific: what is material on one level is a formal bundle of possibilities for the level below; what is formal on one level is material on the level above.

For these reasons, the single step on the ladder of sciences is only partially independent in Peirce, hence also the tendency of his own investigations to zigzag between the levels. His architecture of theories thus forms a sort of phenomenological theory of aspects: the hierarchy of sciences is an architecture of more and less general aspects of the phenomena, not completely independent domains. Finally, Peirce’s realism has as a result a somewhat disturbing style of thinking: many of his central concepts receive many, often highly different determinations which has often led interpreters to assume inconsistencies or theoretical developments in Peirce where none necessarily exist. When Peirce, for instance, determines the icon as the sign possessing a similarity to its object, and elsewhere determines it as the sign by the contemplation of which it is possible to learn more about its object, then they are not conflicting definitions. Peirce’s determinations of concepts are rarely definitions at all in the sense that they provide necessary and sufficient conditions exhausting the phenomenon in question. His determinations should rather be seen as descriptions from different perspectives of a real (and maybe ideal) object – without these descriptions necessarily conflicting. This style of thinking can, however, be seen as motivated by metaphysical continuity. When continuous grading between concepts is the rule, definitions in terms of necessary and sufficient conditions should not be expected to be exhaustive.

Husserl’s Flip-Flop on Arithmetic Axiomatics. Thought of the Day 118.0


Husserl’s position in his Philosophy of Arithmetic (Psychological and Logical Investigations with Supplementary Texts) was resolutely anti-axiomatic. He attacked those who fell into remote, artificial constructions which, with the intent of building the elementary arithmetic concepts out of their ultimate definitional properties, interpret and change their meaning so much that totally strange, practically and scientifically useless conceptual formations finally result. Especially targeted was Frege’s ideal of the

founding of arithmetic on a sequence of formal definitions, out of which all the theorems of that science could be deduced purely syllogistically.

As soon as one comes to the ultimate, elemental concepts, Husserl reasoned, all defining has to come to an end. All one can then do is to point to the concrete phenomena from or through which the concepts are abstracted and show the nature of the abstractive process. A verbal explanation should place us in the proper state of mind for picking out, in inner or outer intuition, the abstract moments intended and for reproducing in ourselves the mental processes required for the formation of the concept. He said that his analyses had shown with incontestable clarity that the concepts of multiplicity and unity rest directly upon ultimate, elemental psychical data, and so belong among the indefinable concepts. Since the concept of number was so closely joined to them, one could scarcely speak of defining it either. All these points are made on the only pages of Philosophy of Arithmetic that Husserl ever explicitly retracted.

In On the Concept of Number, Husserl had set out to anchor arithmetical concepts in direct experience by analyzing the actual psychological processes to which he thought the concept of number owed its genesis. To obtain the concept of number of a concrete set of objects, say A, A, and A, he explained, one abstracts from the particular characteristics of the individual contents collected, only considering and retaining each one insofar as it is a something or a one. Regarding their collective combination, one thus obtains the general form of the set belonging to the set in question: one and one, etc. and. . . and one, to which a number name is assigned.

The enthusiastic espousal of psychologism of On the Concept of Number is not found in Philosophy of Arithmetic. Husserl later confessed that doubts about basic differences between the concept of number and the concept of collecting, which was all that could be obtained from reflection on acts, had troubled and tormented him from the very beginning and had eventually extended to all categorial concepts and to concepts of objectivities of any sort whatsoever, ultimately to include modern analysis and the theory of manifolds, and simultaneously to mathematical logic and the entire field of logic in general. He did not see how one could reconcile the objectivity of mathematics with psychological foundations for logic.

In sharp contrast to Brouwer who denounced logic as a source of truth, from the mid-1890s on, Husserl defended the view, which he attributed to Frege’s teacher Hermann Lotze, that pure arithmetic was basically no more than a branch of logic that had undergone independent development. He bid students not to be “scared” by that thought and to grow used to Lotze’s initially strange idea that arithmetic was only a particularly highly developed piece of logic.

Years later, Husserl would explain in Formal and Transcendental Logic that his

war against logical psychologism was meant to serve no other end than the supremely important one of making the specific province of analytic logic visible in its purity and ideal particularity, freeing it from the psychologizing confusions and misinterpretations in which it had remained enmeshed from the beginning.

He had come to see arithmetic truths as being analytic, as grounded in meanings independently of matters of fact. He had come to believe that the entire overthrowing of psychologism through phenomenology showed that his analyses in On the Concept of Number and Philosophy of Arithmetic had to be considered a pure a priori analysis of essence. For him, pure arithmetic, pure mathematics, and pure logic were a priori disciplines entirely grounded in conceptual essentialities, where truth was nothing other than the analysis of essences or concepts. Pure mathematics as pure arithmetic investigated what is grounded in the essence of number. Pure mathematical laws were laws of essence.

He is said to have told his students that it was to be stressed repeatedly and emphatically that the ideal entities so unpleasant for empiricistic logic, and so consistently disregarded by it, had not been artificially devised either by himself, or by Bolzano, but were given beforehand by the meaning of the universal talk of propositions and truths indispensable in all the sciences. This, he said, was an indubitable fact that had to be the starting point of all logic. All purely mathematical propositions, he taught, express something about the essence of what is mathematical. Their denial is consequently an absurdity. Denying a proposition of the natural sciences, a proposition about real matters of fact, never means an absurdity, a contradiction in terms. In denying the law of gravity, I cast experience to the wind. I violate the evident, extremely valuable probability that experience has established for the laws. But, I do not say anything “unthinkable,” absurd, something that nullifies the meaning of the word as I do when I say that 2 × 2 is not 4, but 5.

Husserl taught that every judgment either is a truth or cannot be a truth, that every presentation either accorded with a possible experience adequately redeeming it, or was in conflict with the experience, and that grounded in the essence of agreement was the fact that it was incompatible with the conflict, and grounded in the essence of conflict that it was incompatible with agreement. For him, that meant that truth ruled out falsehood and falsehood ruled out truth. And, likewise, existence and non-existence, correctness and incorrectness cancelled one another out in every sense. He believed that that became immediately apparent as soon as one had clarified the essence of existence and truth, of correctness and incorrectness, of Evidenz as consciousness of givenness, of being and not-being in fully redeeming intuition.

At the same time, Husserl contended, one grasps the “ultimate meaning” of the basic logical law of contradiction and of the excluded middle. When we state the law of validity that of any two contradictory propositions one holds and the other does not hold, when we say that for every proposition there is a contradictory one, Husserl explained, then we are continually speaking of the proposition in its ideal unity and not at all about mental experiences of individuals, not even in the most general way. With talk of truth it is always a matter of propositions in their ideal unity, of the meaning of statements, a matter of something identical and atemporal. What lies in the identically-ideal meaning of one’s words, what one cannot deny without invalidating the fixed meaning of one’s words has nothing at all to do with experience and induction. It has only to do with concepts. In sharp contrast to this, Brouwer saw intuitionistic mathematics as deviating from classical mathematics because the latter uses logic to generate theorems and in particular applies the principle of the excluded middle. He believed that Intuitionism had proven that no mathematical reality corresponds to the affirmation of the principle of the excluded middle and to conclusions derived by means of it. He reasoned that “since logic is based on mathematics – and not vice versa – the use of the Principle of the Excluded Middle is not permissible as part of a mathematical proof.”

Epistemological Constraints to Finitism. Thought of the Day 68.0


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.

Conjuncted: Axiomatizing Artificial Intelligence. Note Quote.


Solomonoff’s work was seminal in that he has single-handedly axiomatized AI, discovering the minimal necessary conditions for any machine to attain general intelligence.

Informally, these axioms are:

AI0 AI must have in its possession a universal computer M (Universality). AI1 AI must be able to learn any solution expressed in M’s code (Learning recursive solutions).
AI2 AI must use probabilistic prediction (Bayes’ theorem).
AI3 AI must embody in its learning a principle of induction (Occam’s razor).

While it may be possible to give a more compact characterization, these are ultimately what is necessary for the kind of general learning that Solomonoff induction achieves. ALP can be seen as a complete formalization of Occam’s razor (as well as Epicurus’s principle)  and thus serve as the foundation of universal induction, capable of solving all AI problems of significance. The axioms are important because they allow us to assess whether a system is capable of general intelligence or not.

Obviously, AI1 entails AI0, therefore AI0 is redundant, and can be omitted entirely, however we stated it separately only for historical reasons, as one of the landmarks of early AI research, in retrospect, was the invention of the universal computer, which goes back to Leibniz’s idea of a universal language (characteristica universalis) that can express every statement in science and mathematics, and has found its perfect embodiment in Turing’s research. A related achievement of early AI was the development of LISP, a universal computer based on lambda calculus (which is a functional model of computation) that has shaped much of early AI research.

Minimum Message Length (MML) principle introduced in 1968 is a formalization of induction developed within the framework of classical information theory, which establishes a trade-off between model complexity and fit-to-data by finding the minimal message that encodes both the model and the data. This trade-off is quite similar to the earlier forms of induction that Solomonoff developed, however independently discovered. Dowe points out that Occam’s razor means choosing the simplest single theory when data is equally matched, which MML formalizes perfectly (and is functional otherwise in the case of inequal fits) while Solomonoff induction maintains a mixture of alternative solutions. However, it was Solomonoff who first observed the importance of universality for AI (AI0-AI1). The plurality of probabilistic approaches to induction supports the importance of AI3 (as well as hinting that diversity of solutions may be useful). AI2, however, does not require much explanation.