Synthetic Philosophy and Deductive Engineering
I am not a historian, nor an evolutionary biologist, or an academic of any kind, but I have taken some interest in the origins and trajectory of rationality, science and engineering. The historical background discussed here has influenced the philosophy and architecture of SPaDE and has contributed to motivating the ways in which the approach being progressed in SPaDE differs from (and perhaps complements) mainstream ASI (Artificial Super-intelligence) research and development.
The current mainstream builds LLMs (Large Language Models) which are thought and talked of as “neural networks”, architecturally inspired (if loosely) by the structure of the human brain. These neural networks begin as tabulae rasae (blank slates) and are trained on large corpora of text (and/or other media), and then used to generate similar media in response to prompts. Though this is a far cry from the way in which the human brain evolved, develops and learns, it is proving very effective once sufficient computational resource is provided, and has substantially reduced estimates of the timescales to AGI, ASI, “The Singularity” (RSI) and sustainable abundance for all (now expected by some key proponents to be achieved within a decade).
Much of the research on AI over the past century has not adopted this approach, but has involved more deliberate architectural thinking about the representation of knowledge and the use of automated reasoning to derive conclusions from that knowledge. Because of its prominence in the earliest research on AI this approach is sometimes called GOFAI, but perhaps more informatively as the “symbolic” or “logic-based” approach. This kind of research has been marginalised by the technical and commercial success of the “connectionist” (neural net) approach, at least in the industrial progression of AI if not the more diverse academic research, and has not been prominent in the recent commercial development of AGI or ASI.
Alongside that main thrust, there have been some impressive results achieved by means much closer to the symbolic approach, notably Deep Mind’s AlphaZero and AlphaFold, which use neural nets for heuristics in MCTS (Monte Carlo Tree Search) and reinforcement learning. These provide a general model for the construction of intelligent systems specialised in those domains constituting perfect information spaces, and can provide core competence for a broader range of applications related to such spaces.
In Mathematics both informal and formal proofs have been generated by neural nets, and the formal proofs have been verified by proof assistants. There does not appear to be any work which has achieved prominence by more systematic use of symbolic methods of the kind envisaged in SPaDE.
SPaDE advocates for and seeks, in pursuit of its goals, to contribute to the adoption of more systematic and rigorous approaches to the representation of knowledge and the use of deductive reasoning in the development of ASI. It is not predicated on the belief that such approaches are essential to achieving ASI, or sustainable abundance, but in the expectation that eventually ASI will see the benefits of adopting such methods and will seek to exploit formal declarative knowledge representation and deductive reasoning in its own development and operation. Because this is not yet done by homo sapiens, the training of ASI does not encompass effective methods along the lines proposed by SPaDE or any other approach to the same objective. But the techniques proposed are mostly in the literature, and the impediments to their full exploitation by homo sapiens are not significant to ASI, which can therefore be expected to adopt them as soon as it assumes architectural responsibilities in the engineering of its successors.
The remainder of this document falls into the following sections:
This is primarily a story of cultural evolution advancing epistemology, knowledge representation and deductive reason.
The following waypoints are of interest:
Though, from a modern perspective, declarative knowledge and deductive inference are linguistic (symbolic) phenomena, the distinction between declarative and procedural memory substantially predates the evolution of language.
Procedural memory goes back to the earliest organisms, and is implicated in the evolution of the first nervous systems.
Declarative memory is first inferred from consideration of the capabilities of modern organisms, locating those capabilities in cerebral structures (such as the hippocampus) which first emerged in early vertebrates.
Reasoning capabilities are implicated when behaviours are observed which are not explicable by simple stimulus-response mechanisms, and which appear to be based on the use of declarative knowledge to derive conclusions about the world which are then used to guide behaviour. It is premature at this stage to distinguish deductive reasoning, inductive reasoning and abductive reasoning. One reason for this is that the representation of declarative knowledge as memory lacks a definite semantics.
In the context of a long history of declarative memory (in various forms), the ability to communicate the knowledge stored in declarative memory is a major evolutionary advantage, and the emergence of declarative language is a major evolutionary milestone.
It is generally believed that language co-evolved with the development of the human brain, and that the emergence of language was a major factor in the development of human intelligence. It is therefore reasonable to date the origin of language to the emergence of anatomically modern humans, and to locate that emergence in Africa around 300,000 years ago. Of that, an important part is declarative language, the means to communicate declarative knowledge, sharing the content of declarative memory.
To enable communication, the idosyncratic reprsentation of declarative memory in synaptic connections and weights must be translated into a common oral representation. The effectiveness of communication then depends on common understanding of the semantics of this shared representation, though linguistic diversity at various levels may work against this.
Competence in a language necessarily involves knowledge of conceptual inclusions, and hence relations of entailment underpinning deductive inference.
As culuture evolves, at first very slowly indeed, language develops with it. A point of inflection in that development is the emergence of written language.
Agriculture and early civilisations placed demands for administration, trade, taxation, land measurement, and construction which stimulated the development of mathematics in the form of arithmetic and geometric techniques for calculation and measurement, which depend upon written notations.
These early written forms elaborate within half a millenium into written language capable of capturing bulk of the oral tradition, contributing greatly to the preservation and propagation of cultural knowledge, in written declarative language.
We are now 90% through the time from the advent of anatomically modern humans to the present day, and we have just learned to write. All the achievements of Mathematics, Science and Engineering which have transformed the world, and which are now being used to transform the future, took place in that remaining 1%.
The transformation of mathematics into a theoretical discipline takes place early in classical Greece, with the development of axiomatic method.
This makes deductive reasoning more explicit by requiring that each theory is derived from a fixed set of premises of various kinds (axioms, postulates, definitions, …) by chains of reason in which each theorem is inferred exclusively from these presmises or previous theorems. Though the reasoning in fact is deductive, the rules of inference are not explicitly defined, and it will not be for over two millennia that logical systems formalising the rules of inference are developed.
After around 300 years of development of axiomatic mathematics, Aristotle undertook the first study of logic in the collection of books which are now known as the Organon.
An important theme in those books is the idea of demonstrative science, which aims to expand the deductive methods adopted in mathematics to the whole of science as it was then known.
A part of this is his study of the syllogism, which is a systematic treatment of certain aspects of deductive reasoning. Aristotles influence on the development of logic and science was profound, and lasted for over a millennium, until the Renaissance and the Scientific Revolution. But the syllogism became a straitjacket, falling short of the needs of mathematics and science, and inhibiting the development of more flexible and powerful methods of reasoning.
This was not an impediment to the continued advancement of those disciplines, in which the deductive reasoning remained informal but effective.
In transiting from the use of deduction in mathematics to its use in science, it is the identification of the premises from which deduction may proceed which is problematic, and it is in this that modern science was to be transformative.
This transitional period taking us from late antiquity through to the renaissance is primarily significant for its transmission of the work of Aristotle and the stoics as a base on which the later work of Leibniz could build.
Perhaps because of the successes of the axiomatic method in Greek mathematics, Aristotle’s demonstrative science says more about how first principles are applied to derive conclusions than about how those first principles are identified. The modern scientific method, emerging in the Renaissance and Scientific Revolution, is more concerned with the identification and verification of first principles, and the role of deduction in that process.
This leaves the importance of deduction to science untouched (despite scepticism about value of Aristotelian syllogistic logic), within rather more complex and evolving conceptions of scientific method.
A key factor in the new science is the adoption of numerical methods, and the focus on what Aristotle called “efficient causes” (what determines outcomes) rather than “final causes” (the purpose realised). These factors are largely orthogonal to the evolving role of deduction.
Deduction becomes involved in the identification of first principles from observed phenomena as well as in their application to explanation or prediction of phenomena. The methods are described by extension of the Greek terminology for the methods of proof or construction as “analytic” (from the phenomena to the principles) and “synthetic” (from the principles to the phenomena). These methods correspond loosely to the methods later to be characterised as “deductive-nomological” (reasoning from laws to phenomena) and “hypothetico-deductive” (reasoning from hypotheses to phenomena), in which the latter serves primarily in establishing laws and the former in applying them.
The emphasis in early modern science on the roles of deduction and induction was debated, and gave rise to the later characterisation of philosophers as “rationalists” and “empiricists”. Of the rationalists who emphasise the role of deduction, Descartes came closest to the Aristotelian demonstrative science, but the most enthusiastic proponent of Aristotle’s logic is found in Leibniz, who also developed the idea of a “universal characteristic” and a “calculus ratiocinator” which would enable all reasoning to be reduced to calculation. Leibniz’s ideas were not taken up in his own time, but were rediscovered and developed in the 19th century by Boole, Frege and Peano, and have been influential in the development of modern logic, mathematics, computer science and artificial intelligence.
A simple identification of Frege’s Begriffsschrift with Leibniz’s calculus ratiocinator is flawed. Frege himself insisted that he aimed at a lingua characterica (a language that expresses content), not a mere calculus of mechanical inference, and he criticised Boolean algebra for being only the latter. Nor is the Begriffsschrift the full universal characteristic Leibniz envisaged, which was meant to cover the content of all the sciences. Frege presents a more limited, stepwise contribution: a subject-matter-neutral language of pure thought, of which the logicist reduction of arithmetic is the first intended application.
The main factor in the development of logic in the 19th century was the engagement of mathematics with logic, which seems to have had three principal stimuli.
The first was that mathematics became less exclusively focussed on number and geometry, and more open to diverse forms of mathematical structure. From this point of view, the algebraic treatment of propositional logic by Boole and De Morgan was a natural development.
The second was the perceived need for a return to rigor in mathematical analysis, which under the impetus of its applications in science and engineering had mushroomed despite a lack of clarity about its fundamental concepts, notably the real number system and the concept of function, both of which were ontologically novel and opaque. This thread was pursued most effectively by Cauchy, Weierstrass, Dedekind and Cantor, leading to the arithmetisation of analysis and the formal definition of real numbers in terms of Dedekind cuts or Cauchy sequences of rationals.
The third was philosophical. In the eighteenth century a divergence appeared between David Hume and Immanuel Kant. In Hume’s philosophy the division of knowledge into “relations of ideas” and “matters of fact” has a central place and mathematics is placed in the former category. That distinction is an epistemological classification of the objects of human reason, not a psychologistic theory of mental processes in the sense Frege later attacked. Relations of ideas are those propositions whose denial is contradictory and which can be known by the mere operation of thought, independently of experience; matters of fact are contingent and known only through experience. Kant transformed and refined the distinction, splitting the a priori/a posteriori axis from the analytic/synthetic one and introducing synthetic a priori judgments, of which he took mathematics to be the prime example. For Hume, mathematics belonged to the realm of “relations of ideas”, and hence to logic, from which Kant demurred.
The idea that a mathematical discipline should rest on a small number of first principles is ancient. Euclid’s Elements is the paradigm, and for two millennia it served as the model of rigorous foundation, though it was a foundation for geometry, not for mathematics as a whole. The modern project of providing a single foundation for the whole of mathematics is a product of the nineteenth century: the arithmetisation of analysis (eliminating geometric intuition from the calculus), the rise of set theory as a general language of collections, the discovery of non-Euclidean geometries (undermining the self-evidence of Euclid), and finally the logical and set-theoretic paradoxes, which made an explicit secure foundation urgent.
Set-theoretic grounding (Dedekind, Cantor) and logic-plus-definitions (Frege, and in a different style Peano) developed in parallel and often overlapped. “Logic” still carried traditional connotations, and early set theory was not yet sharply distinguished from logic. The clean opposition of two rival programmes — logicism versus axiomatic set theory — crystallised around 1900–1910, once Russell had formulated an explicit logicist thesis, Zermelo had axiomatised set theory, and the paradoxes had forced a choice between different ways of restricting comprehension. Hilbert was already explicit, in the 1900 Paris lecture, about the foundational importance of set theory (the first problem is the continuum hypothesis; the second is the consistency of arithmetic, closely tied in his mind to analysis and set theory). The detailed programme of formalisation and finitary consistency proof is developed more fully in the following two decades.
In a context in which mathematics had been progressing a profound foundational re-construction, Gottlob Frege set out to refute Kant by showing that mathematics could be reduced to logic. The conception of logic which he tabled for that purpose was Begriffsschrift (concept notation). It is clear from the title and the Preface that this was intended, from the outset, as a general logical language for pure thought, independent of any particular subject-matter, not a notation confined to mathematics. The immediate motive is the rigorous analysis of mathematical inference; the intended scope is the laws of pure thought as such. Frege was not engaging with Cantorian set theory when he published the Begriffsschrift in 1879; that contact begins in the Grundlagen der Arithmetik (1884). Two aspects of his ambition were of particular importance.
Frege was the first to attempt not merely the provision of a small set of principles (axioms) from which inference should begin, but also to define precisely the rules of inference themselves governing the derivation of new truths from those axioms.
Frege enunciated the motto: mathematics = logic + definitions
that is, that all mathematical truths could be derived deductively from definitions of mathematical concepts.
The latter conception was controversial but influential, and is the substance of the position in the philosophy of logic which came to be called logicism. At this stage the significance and difficulty of ontology was not fully appreciated, and Frege’s logical foundations for mathematics were found to be inconsistent for lack of a clear underlying ontology. The principal attack on that conception of mathematics and its foundations was connected with the difficulties associated with ontology, and hence with whether those logical systems which were ontologically adequate could properly be considered purely logical.
Nevertheless, Frege’s work was seminal. In 1908, two candidates for “logic” which realised the main elements of Frege’s ambition were on the table: Zermelo’s Axiomatic Set Theory and Russell’s Theory of Types. Neither, as published in 1908, satisfied Frege’s desire for precisely defined rules of inference, but both were subsequently refined to meet that requirement. They were contemporary, parallel attempts at a secure foundation for classical mathematics after the paradoxes, and in that broad practical sense more-or-less equivalent. Philosophically, structurally, and in their later influence they diverge.
Russell’s “Mathematical Logic as Based on the Theory of Types” is explicitly logicist. It is the first systematic published presentation of the ramified theory of types (types and orders, motivated by the vicious-circle principle). A precursor of the Axiom of Reducibility appears there; the axiom is fully articulated, named, and relied upon only in Principia Mathematica (1910–13), where it is needed to recover enough impredicativity for classical mathematics.
Zermelo’s “Untersuchungen über die Grundlagen der Mengenlehre I” is not logicist and is not a formal system in the Hilbertian sense. It is an informal mathematical axiomatization of set theory, in the spirit of Hilbert’s axiomatic method as exemplified in the Grundlagen der Geometrie. Zermelo isolates a small number of principles (Extensionality, Elementary Sets, Separation, Power Set, Union, Infinity, Choice) strong enough to retain Cantor’s theory and the well-ordering theorem, and restrictive enough to block the known paradoxes. There is no type hierarchy; sets are formed iteratively. The presentation has no precise formal language or fully formalized rules of inference. Zermelo notes the desirability of a consistency proof, again in the Hilbertian spirit, without supplying one. His motivation is pragmatic and mathematical rather than belonging to any of the classic “isms” (logicism, formalism, intuitionism): secure Cantorian set theory, and give working mathematicians a reliable toolkit.
In the longer run they are not equivalent. Zermelo’s system, refined into ZF and ZFC by Fraenkel, Skolem and others, proved more flexible and closer to ordinary mathematical practice, and became the dominant foundation. Russell’s ramified hierarchy is more restrictive; the Axiom of Reducibility was widely seen as a weakness. Simplified type theory survived, and is the line from which Church’s Simple Theory of Types, and hence the logic of SPaDE, descends.
These two systems are therefore prototypes for the logical foundation systems which are most important in this context, and we may identify more specifically the later derivatives of these two systems in the first order theory known as Zermelo-Fraenkel Set Theory with Choice (ZFC), and the higher order theory known as Church’s Simple Theory of Types (STT), augmented primarily by polymorphism in the logic of Cambridge HOL.
An important development here is the emergence of meta-mathematics, the study of mathematics itself as a mathematical subject. This may be said to have begun in three stages: non-syntactic reasoning about axiom systems, non-arithmetic reasoning about syntax, and arithmetisation.
This first stage is also the occasion of the Frege–Hilbert controversy (the 1899–1900 correspondence, and Frege’s essays of 1903 and 1906). Frege held that axioms are true thoughts with fixed sense and reference; consistency follows from their truth; definitions fix meanings, axioms assert truths. Hilbert treated axioms as implicit definitions of the primitive terms, and took consistency as the criterion of mathematical existence: if the arbitrarily given axioms do not contradict one another with all their consequences, they are true and the things they define exist. Frege objected that by reinterpreting the primitives one changes the subject-matter, so model constructions do not address the original geometric thoughts. Hilbert largely stopped responding after 1900. His more abstract, structural, model-oriented conception prevailed and became foundational for modern mathematics and model theory.
The contrast is sometimes described as that between a “universalist” conception of logic (Frege: one all-encompassing language of thought, no external standpoint from which to study the system metatheoretically) and a “pluralist” or schematic one (Hilbert: formal systems that admit multiple interpretations and can be studied from outside). That framing does not appear in the correspondence itself. It is a later historiographical construct, first clearly articulated by Jean van Heijenoort in 1967 (“Logic as Calculus and Logic as Language”).
Hilbert and Ackermann’s Grundzüge der theoretischen Logik (1928) belongs here. It reasons about the syntax of first-order logic and poses the completeness of that system as an open problem; it does not prove it. Gödel settled that problem positively in his 1929 dissertation.
The completeness problem that incompleteness answered is a different one. Hilbert posed it publicly in “Probleme der Grundlegung der Mathematik” at the International Congress of Mathematicians in Bologna, September 1928 (published 1929): prove that the axiom system for number theory is formally complete (every sentence or its negation is provable).
Thus: the consistency problem of 1900 helped motivate the metamathematical enterprise; the more precise completeness problems of the late 1920s, together with ongoing consistency efforts, provoked the reduction of metamathematics to arithmetic. That reduction was to be influential in Carnap’s formalisation of science, and seminal for later developments in artificial intelligence such as recursive self-improvement and the singularity.
Logical Positivism was that manifestation of positivism created in the wake of these advances in the logical foundations of mathematics. In the philosophy of Rudolf Carnap, inspired by the work of Bertrand Russell on the formalisation of mathematics and his ideas for “scientific philosophy”, it promoted and facilitated the application of formal deductive systems to the advancement of logical rigour in philosophy and science.
Carnap remained deeply influenced by Frege (whose lectures he had attended) and by Russellian logicism. In the 1920s he still largely operated within a framework that treated mathematics as reducible to pure logic (“mathematics = logic + definitions”). The need to formalise the empirical sciences made that formula inadequate: scientific languages require descriptive vocabulary and rules that cannot be reduced to purely logical definitions.
The Logical Syntax of Language (1934; English 1937) is the decisive document. Carnap presents it as a synthesis of logicism and formalism (§84): mathematical truths remain analytic (without empirical content), while Hilbert’s syntactic methods and the freedom to construct multiple systems are adopted. Gödel’s arithmetisation supplies the technical means by which the syntax of a language can itself be formalised and studied mathematically, often inside an arithmetical metalanguage. The Principle of Tolerance states that there are no “morals” in logic: one may construct any formal language one likes, provided the rules are stated clearly. Different sciences, or different purposes, may adopt different linguistic frameworks, each with its own logical and mathematical apparatus. This is a step away from Fregean universalism toward Hilbert, and it is already implicit in the universalist/pluralist contrast later named by van Heijenoort.
Carnap’s analytic statements are, in scope, a formal successor of Hume’s relations of ideas: a domain of truths fixed by the rules of a language rather than by matters of fact. The Vienna Circle more often cited Wittgenstein, Russell and Frege than Hume, but the structural parallel is real.
Synthetic Philosophy, as developed in connection with SPaDE, is intended as a successor to Logical Positivism, oriented more broadly to Science, Technology, Engineering and Mathematics (STEM) and to all applications in which deduction may play a role. It inherits from Carnap the freedom to construct frameworks, and from the Frege–Russell line the demand for a single underlying representation of declarative knowledge. The two are reconciled, as they must be if a distributed repository is to serve many sciences, by a preferred foundational institution into which other declarative languages may be semantically embedded.
Computers, computer science and artificial intelligence would eventually transform the practical application of deduction, but en route the machinery necessary to settle important issues in metamathematics would impact the program of Rudolf Carnap in relation to the formalisation of science, and pave the way for some key features of the mechanisation of deduction and the evolution of the concept of deduction itself.
This is a summary of the key elements of the understanding of SPaDE about the nature of declarative knowledge and deduction and their relevance to SPaDE.
Declarative knowledge is a domain within which it is possible to realise high degrees of precision in semantics and confidence in truth.