xi

philosophy of mathematics and physics

[xi @a_simple_theory]

the physical foundations of abstract mathematics

1/

#numbertheory #maths #physics

[xi @a_simple_theory]

FUN FACT

the entirety of physics relies on implicitly accepting that basic rules of quantity and combination are natural laws

#numbertheory #maths #physics

[Fermat's Library @fermatslibrary]

FUN FACT

During a lecture Lord Kelvin wrote an integral on the board, turned to the class and asked if they knew what a mathematician is.

"A mathematician is one to whom that is as obvious as that twice two makes four is to you. Liouville was a mathematician."

[xi @a_simple_theory]

physical objects can be functionally identical

[xi @a_simple_theory]

what we call numbers are natural constants

#numbertheory #maths #physics

[xi @a_simple_theory]

the set of rules and operations and patterns we call mathematics

#numbertheory #maths #physics

[xi @a_simple_theory]

the patterns which emerge from increasing the complexity of a singularity of existence through the recursive application of natural first principles

#numbertheory #maths #physics

[xi @a_simple_theory]

empirical origins of mathematical law

#numbertheory #maths #physics

https://t.co/1ETBjRWI8k

[xi @a_simple_theory]

maths 'worked' for 1000s of years before the turn to abstraction and the Peano axioms

#numbertheory #maths #physics

[xi @a_simple_theory]

the question is not IF it is a 'mathematical' universe - because of course it fucking is - the question is WHAT SORT of mathematical universe

#numbertheory #maths #physics

[xi @a_simple_theory]

the beauty of mathematics is shown to anyone who consciously experiences physical reality

#numbertheory #maths #physics

[Physics In History @PhysInHistory]

“The beauty of mathematics only shows itself to more patient followers.”

- Maryam Mirzakhani

[xi @a_simple_theory]

the two frameworks modern physics rests on, both rely on the same fundamental principles

this is not a coincidence
this is not insignificant

basic 'mathematical' rules of quantity and combination are laws of physics

[Philosophy Of Physics @PhilosophyOfPhy]

Modern physics rests on two extraordinarily successful frameworks that describe nature in very different ways.

Quantum mechanics describes particles, fields and their interactions at microscopic scales. General relativity describes gravity not as an ordinary force, but as the curvature of spacetime produced by matter and energy.

Both theories work remarkably well in the domains where we have tested them. The difficulty appears when both should matter at the same time, such as in the very early universe or in regions where spacetime curvature becomes extreme.

There is an important qualification. General relativity can already be treated as an effective quantum field theory at sufficiently low energies. Physicists can calculate small quantum corrections to gravitational interactions this way. What remains unknown is the deeper description that continues to work at arbitrarily high energies, where the usual treatment of Einstein's theory stops being sufficient.

Several very different ideas have grown from this problem.

String theory changes the starting point. Instead of assuming that elementary particles are mathematical points, it describes fundamental objects as tiny one-dimensional strings. Different vibrational states appear as different particles. One remarkable result is that the spectrum of a closed quantum string naturally contains a massless spin-two state with the properties expected of the hypothetical graviton. Gravity therefore appears inside the theory rather than being inserted afterward. String theory has developed into a powerful mathematical framework, but we still have no experimental evidence showing that fundamental strings describe nature.

Loop quantum gravity takes almost the opposite route. Instead of replacing particles with strings, it tries to quantize Einstein's gravitational field itself. In this framework, geometry becomes quantum mechanical. Quantities such as area and volume are represented by quantum operators with discrete spectra. Smooth spacetime would then be an approximation to a deeper quantum geometry. The mathematical structure is well developed, but connecting it completely to the familiar low-energy world and finding decisive experimental evidence remain major challenges.

A third possibility is asymptotic safety. Perhaps gravity does not need entirely new fundamental objects. The problem might instead be solved if the behavior of gravitational interactions approaches a special renormalization-group fixed point at extremely high energies. If such a fixed point governs the theory, gravity could remain predictive even where ordinary perturbative calculations fail. Considerable theoretical evidence has been found for this possibility, but it has not been experimentally established.

Other approaches question whether smooth spacetime exists fundamentally at all.

In causal dynamical triangulations, physicists construct quantum spacetime from simple geometric building blocks and sum over many possible spacetime configurations. Large-scale, approximately four-dimensional geometry can emerge from these microscopic structures in numerical calculations.

Causal set theory goes further and proposes that the basic structure of spacetime may consist of discrete events connected only by causal relations. Continuous geometry would then appear only when the system is viewed on sufficiently large scales.

And holography offers perhaps the most radical change of perspective.

The holographic principle, especially through AdS/CFT duality, shows that in certain theoretical settings a gravitational theory in one spacetime can be equivalent to a quantum theory without gravity defined in fewer dimensions. Research following this discovery has revealed deep connections between quantum entanglement and spacetime geometry.

This has encouraged a striking possibility. Perhaps spacetime is not one of the fundamental ingredients of nature.

Perhaps it emerges from something quantum underneath.

None of these ideas has yet won through experimental confirmation. String theory, loop quantum gravity, asymptotic safety, causal approaches and holography should therefore not be described as established descriptions of our universe. They are different attempts to answer a question for which nature has not yet given us a decisive experimental clue.

But the situation is changing.

For decades, quantum gravity seemed almost inaccessible experimentally because the natural scale associated with it is extraordinarily far beyond existing particle accelerators. Recently, researchers have begun exploring another route. Instead of reaching enormous energies, experiments may place increasingly massive objects into genuinely quantum states and investigate how gravity behaves between them.

Several proposals ask whether gravity can generate entanglement between two quantum systems. Under appropriate assumptions, observing such gravitationally mediated entanglement could tell us that an entirely classical description of the gravitational interaction is insufficient.

These experiments are extremely difficult, and their interpretation is still being debated. They have not detected quantum gravity.

Yet they represent an important shift.

The search is slowly moving from asking only which mathematical theory of quantum gravity is most compelling toward asking what experiment could tell us something fundamental about gravity's quantum nature.

And there is still another possibility worth keeping open.

Perhaps the final theory will not look like string theory, loop quantum gravity, or any framework we currently know. Perhaps gravity is emergent. Perhaps quantum mechanics itself requires modification when gravity becomes important. Or perhaps the concepts of space and time that appear fundamental to us will turn out to be collective phenomena arising from something deeper.

For now, the central question remains unresolved.

Quantum mechanics tells us that physical systems can exist in superpositions and become entangled.

General relativity tells us that gravity is spacetime geometry.

So what happens when spacetime itself must obey the rules of quantum physics?

We still do not know.

That is the problem of quantum gravity.

[xi @a_simple_theory]

the unreasonable effectiveness of mathematics in the natural sciences

and it's already been solved, i'm just waiting for the rest of you to catch up

#numbertheory #maths #physics

[xi @a_simple_theory]

and ALL of this relies on basic combinatorial rules of quantity and magnitude being natural laws of #physics

[xi @a_simple_theory]

the physical foundations of abstract mathematics

1/

#numbertheory #maths #physics

[xi @a_simple_theory]

and ALL of this relies on basic combinatorial rules of quantity and magnitude being natural laws of #physics

[Philosophy Of Physics @PhilosophyOfPhy]

The inverse-square relationship became central to seventeenth-century physics through the study of light and planetary motion.

Isaac Newton later made it fundamental to universal gravitation in the Principia of 1687, showing that gravitational attraction decreases with the square of the distance between bodies.

For a point source radiating equally in every direction, its power spreads across an expanding sphere. Since the sphere’s area is 4πr², the intensity at distance r is

I = P/4πr²

Doubling the distance spreads the same power over four times the area, reducing the intensity to one quarter. At three times the distance, it falls to one ninth.

This simple geometric law explains how light and sound weaken with distance and why gravitational and electric fields follow similar distance dependence. It applies most directly to an ideal point source radiating uniformly through space, without significant absorption or obstruction.

[xi @a_simple_theory]

all this effort just to confirm for the gazillionth time that basic rules of quantity and combination are the most fundamental laws of physics

#numbertheory #maths #physics

[xi @a_simple_theory]

the unreasonable effectiveness of mathematics in the natural sciences

and it's already been solved, i'm just waiting for the rest of you to catch up

#numbertheory #maths #physics

[Physics In History @PhysInHistory]

What's a scientific mystery you believe could be solved in your lifetime?✍️

[xi @a_simple_theory]

the law of conservation of energy is directly equivalent of 1+2=3

#numbertheory #maths #physics

[Philosophy Of Physics @PhilosophyOfPhy]

In 1924, Niels Bohr, Hendrik Kramers, and John Slater proposed that energy and momentum might not be conserved in every individual interaction between radiation and matter.

They suggested that the conservation laws might hold only statistically across many events.

Walther Bothe and Hans Geiger designed an experiment to ask a remarkably direct question: when an X-ray photon scatters from an electron, do the scattered photon and recoiling electron appear together as one physical event?

Using separate detectors and an early coincidence-counting method, they found that the signals occurred together. This supported event-by-event conservation and contradicted the Bohr–Kramers–Slater proposal.

Bothe later described the experiment as asking whether a scattered quantum and recoil electron were simultaneously produced or merely statistically associated.

The technique became far more important than this single debate.

Coincidence and anti-coincidence detection later became foundational tools in cosmic-ray, nuclear, particle, and neutrino experiments. Bothe received half of the 1954 Nobel Prize in Physics for the coincidence method and the discoveries made with it.

( 📷 Bothe explaining the cyclotron to the mayor of Heidelberg, Carl Neinhaus, at the Max Planck Institute for Medical Research, 1955.)

[xi @a_simple_theory]

all this effort just to confirm for the gazillionth time that basic rules of quantity and combination are the most fundamental laws of physics

#numbertheory #maths #physics

[Brian Roemmele @BrianRoemmele]

The Well Just Dropped: 15 Terabytes of Pure Physics Gold Is Now Open Source

The scientific AI world just got a massive upgrade.Polymathic AI, in collaboration with the Flatiron Institute and researchers from Princeton, Cambridge, NYU, Berkeley, Los Alamos, and more, has released The Well: a staggering 15TB collection of high-fidelity physics simulations.

This isn’t toy data.

These are real, expensive-to-run simulations across 16 different physical domains, including turbulent fluid dynamics, supernova explosions, magneto-hydrodynamic cosmic flows, acoustic scattering, and active biological matter.

Until now, reproducing this level of data required weeks on national supercomputers and grant money most teams will never see. The Well changes everything. It’s purpose-built for training PDE surrogate models the AI systems that can replace slow, costly physics solvers with a single fast neural network forward pass.

Everything is fully open source, easy to load with PyTorch, and ready to drop straight into your training pipeline. Researchers and builders can now train on world-class physics data without the insane compute barriers that used to stand in the way.

This is more than just another dataset drop. It’s a serious accelerator for scientific machine learning.The future of physics-informed AI just got a whole lot more accessible.Get it here:
https://t.co/mIW0xA6RAs

[xi @a_simple_theory]

the question is not IF it is a 'mathematical' universe - because of course it fucking is - the question is WHAT SORT of mathematical universe

#numbertheory #maths #physics

[American Physical Society @APSphysics]

Complex numbers have long seemed essential to #QuantumMechanics — but a paper in @PhysRevLett argues they may not be. By reformulating quantum theory using only real numbers, the authors show it can make the same experimentally-testable predictions: https://t.co/RXjoNDr6ti