As strange as it may seem, developments in quantum physics and computer science promise to have important consequences for mathematicians, and for mathematics in general. The idea of what counts as a proof is at issue, as is the future role of human mathematicians. Additionally, some famous intractable problems may be solvable, based on the weirdness of quantum particles.
As strange as it may seem, developments in quantum physics and computer science promise to have important consequences for mathematicians, and for mathematics in general. The idea of what counts as a proof is at issue, as is the future role of human mathematicians. Additionally, some famous intractable problems may be solvable, based on the weirdness of quantum particles.
Quantum computing represents one of the most profound technological and intellectual advances of the twenty-first century. Rooted in the principles of quantum mechanics and enriched by developments in computer science, mathematics, and engineering, quantum computing offers fundamentally new ways of processing information. By exploiting phenomena such as superposition, entanglement, and quantum interference, quantum computers promise to solve certain classes of problems that remain intractable for even the most powerful classical machines.
We will encourage you to develop the three great virtues of a programmer: laziness, impatience, and hubris. – Larry Wall, Programming Perl, 1st edition So there you have it. Hundreds of pages and more than 3000 diagrams later, we've told you (pretty much) everything we know about quantum theory and diagrammatic reasoning. So, where to now? Is it time for everyone to start covering blackboards and filling papers with diagrams? Of course! But even better, what if someone else did all the diagrammatic proving for you while you sit back, relax, and have a beer? That's even better! The fact that the diagrams we use are essentially made up of a finite number of ingredients (namely, spiders) means they are particularly well suited to automated reasoning . In this subfield of artificial intelligence, one develops software that allows a computer to do a whole range of things often associated with human mathematicians: from simply checking mathematical proofs for correctness to automatically searching for new proofs or even new and interesting conjectures to then try and prove automatically. In the past, automated reasoning has typically concerned traditional, formula-based mathematics built on formal logics and set-based algebraic structures, and there it has been very successful. It has provided us with tools called proof assistants , which allow us to automatically construct proofs of mind-bending results, such as Gödel's incompleteness theorems, and rigorously check proofs that are way too big for a human mathematician to get totally correct, like Kepler's conjecture, the four-colour theorem, and the Feit– Thompson theorem (famously massive proofs in geometry, graph theory, and group theory, respectively). In addition to serving essentially as ‘robot teaching assistants’, which tirelessly check the work of human mathematicians, techniques from automated reasoning can actually tell us something new, via conjecture synthesis . Much as a human mathematician would discover features and behaviours of an unfamiliar mathematical creature by ‘poking at it’ (i.e. making educated guesses about how it will behave and trying to prove them), there exist automated techniques that do this at high speed. When it succeeds in a proof, the result is a freshly minted theorem that no human has ever seen or even thought to ask about.
Though the truths of logic and pure mathematics are objective and independent of any contingent facts or laws of nature, our knowledge of these truths depends entirely on our knowledge of the laws of physics. Recent progress in the quantum theory of computation has provided practical instances of this, and forces us to abandon the classical view that computation, and hence mathematical proof, are purely logical notions independent of that of computation as a physical process. Henceforward, a proof must be regarded not as an abstract object or process but as a physical process, a species of computation, whose scope and reliability depend on our knowledge of the physics of the computer concerned.
AI-assisted mathematics is often evaluated on solving predefined problems. In practice, however, many important advances begin earlier, when a vague research intuition is transformed into a concrete problem, a promising route, and a theorem family worth proving. This report studies that stage through a case study that led to sign-embedding quantum algorithms for matrix equations and matrix functions, foundational primitives in quantum linear algebra and operator-output quantum algorithms. The project began with a human-originated intuition that rational approximation is especially effective for jump-type functions such as the sign function, and might therefore serve as a design principle for quantum algorithms. Rather than merely assisting after the problem was fixed, AI-assisted exploration, including workflows later integrated into the agentic AI-mathematician system AIM, played a key role in expanding this intuition into a route map, comparing candidate formulations, and converging toward sign embedding as the central framework. AIM then helped connect a known matrix-sign identity to wider classes of matrix equations and matrix functions, and drafted proof and complexity calculations. The decisive scientific judgments remained human: selecting which human-AI-expanded routes were worth pursuing, rejecting a Cayley-trapezoidal approximation when its validity required a hidden condition, and refining the Sylvester implementation from a coarse quadratic-gap query route to the final factorized and scaled analysis. The report argues that human-AI co-discovery workflows, with systems such as AIM as important components, are most valuable not as standalone theorem provers, but as research partners for problem formation, connection discovery, derivation, and skeptical review inside a human-gated research loop.
While physicists pursue the search for the Grand Unified Theory (GUT) and the expedition to find the Higgs that ought to solidify the Standard Model, philosophers keep inquiring as to what the nature of matter and mind might be and propose various theories. At first sight, the high energy physicist or the philosopher, both human, and the elementary particle seem to have nothing in common, however we contend and argue in this paper that we are looking at different scales of what constitutes universal behaviour and use Herman Dooyeweerd's theory of Modal Aspects to bridge the particle to the mind. To other than a physicist the dual nature of the atom or the so-called elementary particles may be difficult to comprehend and certainly appears unfathomable to intuit. This is due to the fact that the dual nature of matter finds its anchoring within the theoretical and mathematical frameworks provided by quantum mechanics, the latter tends to not be easily accessible to most individual's intuition. What would the consequences be if human nature itself had a dual character? We all know that human nature is both egoistic and altruistic, however it is not often that one considers the fact that egoism is an individual behaviour, while altruism is a collective behaviour. Likewise a particle is an individual manifestation of what we call matter, while wave nature can be considered characteristic of collective behaviour of the same matter.
Recent advances in experimental physics—the queen of reductionist science—aimed at solving the mysteries of the quantum world has led to a deepening of the mystery rather than easing it, dispelling the EPR paradox and pointing out a conceptional error in this now famous gedanken experiment, while lending some authority and validation to quantum mechanics. The strange features of the quantum world are founded on what Bohr dubbed complementarity, or the way a sub-particle entity can behave either as a wave or as a particle; and on Werner Heisenberg's uncertainty principle, which says that a quantum entity such as an electron does not avail itself to a deterministic exact measurement of both position and momentum simultaneously. Such postulates seem to contradict deeply held assumptions by the deterministic and materialistically oriented world of experimental physics where everything, including the atom, can be reduced to a single aspect. There are mounting calls for newer ways of understanding reality that are capable of handling these apparently strange incomprehensible phenomena revealed through the classical observation of the microscopic quantum world. There is also fundamental progress made in bouncing cosmology theories that bridge general relativity to quantum mechanics and provide some insight into the nature of the universe and its transition and pre-history through the big bang.
The thrust of this paper evolves around two pillars. One, is that that the apparent logical fissure emergent in reductionist thought, coming from orthodox science and the application of the scientific method that is manifest in what we know and do not understand in quantum mechanics, can be liaised to our intellectual understanding through the perspective offered by Herman Dooyeweerd's theory of Modal Aspects. The other, is the central and special role humans have in relation to the universe. Through this approach, we will argue that the dual nature of the atom or elementary particles is, rather than being a strange phenomenon, in fact an essential characteristic of what constitutes its nature