What the Scientific Revolution Can Teach Us About the AI Revolution

Closing the feedback loop between mathematics and AI

The scientific revolution didn't happen because physics got better at physics. It happened because physics and mathematics entered into a virtuous cycle — a feedback loop in which each discipline propelled the other forward, unlocking capabilities that neither could have achieved alone. If we look carefully at how that loop worked, it offers a suggestive blueprint for where AI might be headed. Something analogous may be emerging between AI and mathematics today. And if that loop closes tightly, the consequences could be transformative.

The engine of the scientific revolution

Consider the relationship between Newtonian mechanics and calculus. The need to describe motion and forces drove Newton (and Leibniz) to invent calculus. But once calculus existed as a mature mathematical framework, it didn't just serve mechanics — it revolutionised it. Differential equations, dynamical systems, and the elegant reformulations of Lagrangian and Hamiltonian mechanics all followed. Eventually, the mathematical machinery that mechanics had birthed pointed the way toward quantum mechanics, a theory that would have been unthinkable without it. As the tagline goes: mechanics invented calculus; calculus re-invented mechanics.

This pattern repeats across the history of science with striking regularity. Fourier developed his series to solve the heat equation — a physical PDE birthed an entire branch of analysis. That analysis then flowed back to solve countless more physical problems across waves, optics, and communications. Riemannian geometry gave Einstein the language to describe gravity through general relativity; in turn, general relativity spurred the development of differential geometry, which fed back into modelling physical processes like gravitational waves. Noether’s theorem revealed that the conserved quantities physicists had observed — energy, momentum, angular momentum — were consequences of symmetries. That insight fuelled the development of group and representation theory, which now underpins the Standard Model of particle physics.

Figure 1: The feedback loop between mathematics and the sciences: Einstein (general relativity and differential geometry), Noether (symmetries and group theory), Newton (mechanics and calculus), and Fourier (heat equation and harmonic analysis). [Image generated by Gemini.]

In each case, the pattern is the same. A problem in the physical sciences motivates new mathematics. That mathematics matures, develops its own internal logic, and then flows back to transform the original science — and often far beyond it. The feedback loop between mathematics and the physical sciences was, in many ways, the engine of the scientific revolution.

A two-way street

With this historical pattern in mind, now consider AI. The relationship between mathematics and AI is often framed as a one-way street: mathematics provides the tools, AI applies them. But this framing misses what I think is the more interesting and potentially more consequential dynamic — a two-way interaction that mirrors the historical relationship between mathematics and the physical sciences. This is the approach I’ve been taking in much of my research (follow references to our papers for further details).

In one direction, mathematics flows into AI. We build inductive biases into models using symmetries, equivariances, and group theory [1,2,3] (Geometric AI is discussed further in a previous post). We integrate differentiable mathematical modelling — spherical harmonic transforms, physical simulators, geometric structure — directly into learning pipelines [5,6,7,8]. We prove properties of AI models: that certain architectures are stable [3], that certain training procedures converge [9], that certain regularisation strategies recover statistical moments [10] or yield meaningful uncertainties [11]. This is the direction most practitioners think about, and it has delivered remarkable results. Encoding translational symmetry into convolutional neural networks, for instance, was one of the key factors driving the computer vision revolution.

But the other direction — AI flowing into mathematics — is equally fascinating and arguably underappreciated. Deep generative models now serve as powerful priors in inverse problems [12]. Machine learning provides proposal distributions that accelerate statistical inference [13,14,15,16]. Conformal prediction offers a framework for statistically valid uncertainties [17]. Neural emulators act as fast surrogate simulators for expensive physical models [18]. And most visibly, AI is beginning to prove mathematical theorems. In 2024, DeepMind's AlphaProof and AlphaGeometry2 achieved a silver medal at the International Mathematical Olympiad through neuro-symbolic approaches. By 2025, Gemini's deep reasoning achieved gold — with a pure large language model.

Where the loop starts to close

The historical examples above share a common turning point: the moment when the two-way traffic between disciplines becomes a genuine dialogue, each advance in one triggering a response in the other. In AI, the most exciting developments arise precisely where this begins to happen — where the two directions don't just coexist but interact.

Take the trajectory of plug-and-play (PnP) methods in computational imaging (discussed in a previous post). Deep generative models provide powerful learned priors, but naively plugging them into iterative algorithms sacrifices theoretical convergence guarantees and the ability to quantify uncertainty. Mathematics identifies the problem: unconstrained neural networks can violate the Lipschitz conditions needed for convergence. This motivates new mathematical work — constraining the Lipschitz constant of the learned model — which restores convergence [9,11]. In addition, imposing convexity constraints on learned models can provide principled uncertainty quantification [11]. The result is AI that is not just powerful but reliable: interpretable, robust, and backed by theoretical guarantees. The loop here is clear: AI poses a new challenge, mathematics responds, and the resulting synthesis is stronger than either contribution alone.

We see the same dynamic in mathematical theorem proving. The 2024 neuro-symbolic olympiad systems combined neural planning with formal verification in Lean. The 2025 systems leaned more heavily on pure language model reasoning. Now the field is moving toward integrated approaches: LLMs for high-level planning and search, neuro-symbolic systems for proof guarantees. Each generation of AI capability poses new mathematical questions about verification, correctness, and reliability — and the mathematical answers feed back to improve the next generation.

What this achieves

When the loop closes — when AI and mathematics are genuinely in dialogue rather than one merely serving the other — the resulting systems tend to be qualitatively different from those produced by either discipline in isolation. They tend to be more interpretable, because mathematical structure makes the learned representations meaningful. They tend to be more reliable, because theoretical guarantees accompany empirical performance: stable representations through scattering networks, stable training through spectral normalisation, convergence through Lipschitz constraints, meaningful uncertainties through convexity and statistical calibration. And they tend to be more robust, because mathematical modelling and physics-informed design provide a scaffold that pure data-driven approaches lack.

In short, closing the feedback loop doesn't just make AI more mathematical or mathematics more computational. It creates a new kind of capability that is simultaneously more powerful and more trustworthy.

Lessons from history

The scientific revolution was driven by many factors beyond the mathematics-physics feedback loop, and the AI revolution — if that's what we're living through — will be similarly multifaceted. Scaling, data, compute, engineering, and even sociological dynamics around open-source development all play essential roles.

But I do think the historical lesson is worth taking seriously as a lens for thinking about where AI might be headed. The scientific revolution didn't just produce better models of the physical world. It produced a way of producing better models — a methodology that compounded over centuries. The feedback loop between mathematics and physics was central to that methodology.

If something similar is emerging between mathematics and AI, the implications run deep. It suggests that the most transformative advances may not come from scaling alone, or from mathematical theory alone, but from the tight integration of both — from the places where each discipline challenges and extends the other.

Consider the progression of Moravec's paradox in light of this. Early AI excelled at formal reasoning but struggled with perception and motor control — the tasks that feel effortless to humans. Large language models have, in many ways, inverted this: they handle natural language, analogy, and even creative reasoning with remarkable fluency, while struggling with formal mathematical proof. The closing of the AI-mathematics loop might be precisely what resolves this tension, producing systems that combine the intuitive fluency of modern AI with the rigour and reliability that mathematics demands.

Looking ahead

We are, I suspect, still in the early stages of this feedback loop. The mathematical tools being brought to bear on AI — group theory, differential geometry, functional analysis, information theory — are powerful but largely drawn from existing stockpiles. What we haven't yet seen, on any large scale, is AI motivating fundamentally new mathematics in the way that mechanics motivated calculus or general relativity motivated differential geometry. When that starts to happen — when AI doesn't just consume mathematical tools but drives the creation of new ones — the loop will be fully closed.

The scientific revolution teaches us that the most profound advances came not from either mathematics or the physical sciences in isolation, but from the places where they challenged and transformed each other. If AI follows that same pattern, the revolution may be just getting started.

References

[1] Cobb, Wallis, Mavor-Parker, Marignier, Price, d'Avezac, McEwen, Efficient Generalised Spherical CNNs, ICLR (2021), arXiv:2010.11661

[2] Ocampo, Price, McEwen, Scalable and equivariant spherical CNNs by discrete-continuous (DISCO) convolutions, ICLR (2023), arXiv:2209.13603

[3] McEwen, Wallis, Mavor-Parker, Scattering Networks on the Sphere for Scalable and Rotationally Equivariant Spherical CNNs, ICLR (2022), arXiv:2102.02828

[4] Price, McEwen, Differentiable and accelerated spherical harmonic and Wigner transforms, Journal of Computational Physics (2024), arXiv:2311.14670

[5] Price, Polanska, Whitney, McEwen, Differentiable and accelerated wavelet transforms on the sphere and ball, Journal of Computational Physics (2024), arXiv:2402.01282

[6] Mars, Liaudat, Whitney, Betcke, McEwen, Generative imaging for radio interferometry with fast uncertainty quantification (2025), arXiv:2507.21270

[7] Mars, Betcke, McEwen, Learned radio interferometric imaging for varying visibility coverage, RASTI (2025), arXiv:2405.08958

[8] Mars, Betcke, McEwen, Learned Interferometric Imaging for the SPIDER Instrument, RASTI (2023), arXiv:2301.10260

[9] Ryu, Liu, Wang, Chen, Wang, Yin, Plug-and-Play Methods Provably Converge with Properly Trained Denoisers, ICML (2019), arXiv:1905.05406

[10] Bendel, Ahmad, Schniter, A Regularized Conditional GAN for Posterior Sampling in Image Recovery Problems (2022), arXiv:2210.13389

[11] Liaudat, Mars, Price, Pereyra, Betcke, McEwen, Scalable Bayesian uncertainty quantification with data-driven priors for radio interferometric imaging, RASTI (2024), arXiv:2312.00125

[12] McEwen, Liaudat, Price, Cai, Pereyra, Proximal nested sampling with data-driven priors for physical scientists, MaxEnt (2023), arXiv:2307.00056

[13] McEwen, Wallis, Price, Spurio Mancini, Machine learning assisted Bayesian model comparison: learnt harmonic mean estimator, Statistics & Computing (2022), arXiv:2111.12720

[14] Polanska, Price, Piras, Spurio Mancini, McEwen, Learned harmonic mean estimation of the Bayesian evidence with normalizing flows, Open J. Astrophys. (2024), arXiv:2405.05969

[15] Polanska, Wouters, Pang, Wong, McEwen, Accelerated Bayesian parameter estimation and model selection for gravitational waves with normalizing flows, NeurIPS ML4PS Workshop (2024), arXiv:2410.21076

[16] Polanska, McEwen, Learned harmonic mean estimation of the marginal likelihood for multimodal posteriors with flow matching, MaxEnt (2026), arXiv:2601.18683

[17] Angelopoulos, Bates, A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification (2021), arXiv:2107.07511

[18] Piras, Polanska, Spurio Mancini, Price, McEwen, The future of cosmological likelihood-based inference: accelerated high-dimensional parameter estimation and model comparison, Open J. Astrophys. (2024), arXiv:2405.12965

Ideas my own. Writing a collaborative effort with AI.