John Horton Conway. 26 December 1937—11 April 2020

John Conway was without doubt one of the most celebrated British mathematicians of the last half century. He first gained international recognition in 1968 when he constructed the automorphism group of the then recently-discovered Leech lattice, and in so doing discovered three new sporadic simple groups. At around the same time he invented The Game of Life, which brought him to the attention of a much wider audience and led to a cult following of Lifers. He also combined the methods of Cantor and Dedekind for extending number systems to construct what Donald Knuth (ForMemRS 2003) called ‘surreal numbers’, the achievement of which Conway was probably most proud. Throughout his life he continued to make significant contributions to many branches of mathematics, including number theory, logic, algebra, combinatorics and geometry, and in his later years he teamed up with Simon Kochen to produce the Free Will theorem, which asserts that if humans have free will then, in a certain sense, so do elementary particles. In this biographical memoir I attempt to give some idea of the depth and breadth of Conway's contribution to mathematics.

On the nature of Time and Entanglement

The nature of Time is often at the root of the physical debate and possibly sits at the core of General Relativity and Quantum Mechanics frameworks incompatibility. In the context of the Free Will theorem and of a spacetime described through information, we identify in a thick present the only quantum information potential needed to describe evolution. The analysis of undefined causal orders (through a quantum Controlled-NOT gate and the evolution of the information along an imaginary time) allowed us to describe entanglement (both in space position and time order) as the potential related to an open choice and expressed in a CTC, which develops in a non-local imaginary space within the thick present considered.

From the “Free Will Theorems” to the “Choice Ontology” of Quantum Mechanics

If the concept of “free will” is reduced to that of “choice” all physical world share the latter quality. Anyway the “free will” can be distinguished from the “choice”: The “free will” involves implicitly certain preliminary goal, and the choice is only the mean, by which it can be achieved or not by the one who determines the goal. Thus, for example, an electron has always a choice but not free will unlike a human possessing both. Consequently, and paradoxically, the determinism of classical physics is more subjective and more anthropomorphic than the indeterminism of quantum mechanics for the former presupposes certain deterministic goal implicitly following the model of human freewill behavior.The choice is usually linked to very complicated systems such as human brain or society and even often associated with consciousness. In its background, the material world is deterministic and absolutely devoid of choice. However, quantum mechanics introduces the choice in the fundament of physical world, in the only way, in which it can exist: All exists in the “phase transition” of the present between the uncertain future and the well-ordered past. Thus the present is forced to choose in order to be able to transform the coherent state of future into the well-ordering of past. The concept of choice as if suggests that there is one who chooses. However quantum mechanics involves a generalized case of choice, which can be called “subjectless”: There is certain choice, which originates from the transition of the future into the past. Thus that kind of choice is shared of all existing and does not need any subject: It can be considered as a low of nature.There are a few theorems in quantum mechanics directly relevant to the topic: two of them are called “free will theorems” by their authors, Conway and Kochen, and according to them: “Do we really have free will, or, as a few determined folk maintain, is it all an illusion? We don’t know, but will prove in this paper that if indeed there exist any experimenters with a modicum of free will, then elementary particles must have their own share of this valuable commodity” “The import of the free will theorem is that it is not only current quantum theory, but the world itself that is non-deterministic, so that no future theory can return us to a clockwork universe”Those theorems can be considered as a continuation of the so-called theorems about the absence of “hidden variables” in quantum mechanics.

The Code-Theoretic Axiom: The Third Ontology

A logical physical ontology is code theory, wherein reality is neither deterministic nor random. In light of Conway and Kochen’s free will theorem [The free will theorem, Found. Phys. 36(10) (2006) 1441–1473] and strong free will theorem [The strong free will theorem, Not. Am. Math. Soc. 56(2) (2009) 226–232], we discuss the plausibility of a third axiomatic option — geometric language; the code-theoretic axiom. We suggest that freewill choices at the syntactically-free steps of a geometric language of spacetime form the code-theoretic substrate upon which particle and gravitational physics emerge.