Mathematical Truths and Physical Randomness: A Possible Problem with Math as Fundamental to Physics – AI Insights.

A conversation with an AI about randomness and its connection between mathematical truths and physical reality.

AI prompt 01:

Is it possible to create truly random output using only mathematical formulas, without relying on readings from physical processes?

AI response 01:

Generating **truly random output** solely through **mathematical formulas**—without relying on physical processes—is fundamentally **impossible**. The core reason lies in the deterministic nature of mathematical formulas and algorithms, which produce **pseudorandom numbers** that are repeatable and predictable if the initial conditions or seed are known.

### Why Purely Mathematical Formulas Can’t Create True Randomness

– **Determinism of formulas**: Mathematical functions and algorithms, by design, follow precise rules. For any given input (including seed values), they will always produce the same output.
– **Pseudorandom Numbers**: Most generators, such as Linear Congruential Generators or Mersenne Twister, rely on formulas and initial seeds to generate sequences that appear random but are ultimately predetermined and reproducible if the seed is known.
– **Lack of Inherent Unpredictability**: Complete unpredictability in mathematics requires effects that are fundamentally nondeterministic, which pure formulas lack.

### Physical and Quantum Processes for True Randomness

– **Physical Noise**: Genuine randomness arises from physical phenomena like radioactive decay, thermal noise, or quantum effects, which are inherently nondeterministic.
– **Quantum Phenomena**: Recent technologies utilize quantum effects, such as photon polarization or quantum tunneling, which are mathematically proven to be fundamentally unpredictable, to generate truly random numbers.

### Conclusion

While mathematical formulas can be crafted to produce outputs that **appear random**—and are suitable for most practical purposes—they **cannot** generate *true randomness* on their own, since their behavior is entirely deterministic. True randomness depends on unpredictable physical or quantum phenomena, which formulas alone cannot emulate.

AI prompt 02 (follow-up):

why is that? by my understanding, physical processes rely on mathematical forms, eternal truths of math. is it possible that we haven’t discovered the needed mathematical formulas yet, for truly random output.

AI response 02:

Mathematical formulas indeed describe physical processes, but the key reason we can’t get truly random output from formulas alone lies in the difference between **determinism** and **intrinsic randomness**.

Physical processes depend on fundamental laws often expressed mathematically, yet:

– These laws can be **deterministic** or **probabilistic**. Deterministic laws produce fully predictable outcomes given initial conditions, so no true randomness arises just from formulas there.
– However, in **quantum mechanics**, physical processes inherently embody **probabilistic behavior**. For example, when measuring properties like an electron’s position, outcomes are fundamentally unpredictable despite exact math formulas like the wave function describing probabilities.
– This intrinsic unpredictability (true randomness) in quantum events is not due to lack of knowledge or undiscovered formulas but is a core feature of nature as understood today. It’s mathematically described by probability amplitudes and the collapse of wave functions upon measurement.
– Though classical physics and chaos theory show unpredictability from sensitive dependence on initial data (deterministic chaos), this is not true randomness since it is still determined by the initial state.
– The possibility remains open that deeper “hidden variables” could reproduce determinism beneath quantum randomness, but no conclusive evidence exists to supplant the standard probabilistic interpretation.

Thus, while all physical laws are expressed mathematically, **true randomness as witnessed in quantum phenomena is not merely about missing formulas**—it reflects a fundamental limitation in predictability embedded in the structure of physical law itself.

In summary: Mathematical descriptions capture both deterministic and probabilistic aspects of physical reality, but intrinsic randomness arises from quantum nature, not from a gap in mathematical knowledge or formulas.

AI prompt 03 (follow-up):

years ago, while needing truly random input in my research, but being limited with pseudo-random number generator of the pogramming language ‘basic’, i created loops of several pseudo-random outputs, for example one generator running while other generator randomly stopping the first generator to read the output, then using the result in different and parallel loops, etc., in the end using several readings to randomly select from those the final output to use. was the final reading truly random or still not?

AI response 03:

Your approach of combining multiple pseudo-random number generators (PRNGs) through loops and layering outputs to produce a final reading is a known technique to try to enhance randomness quality. However, the outcome is still fundamentally pseudorandom rather than truly random. This is because:

– Each individual PRNG is deterministic based on its initial seed; combining several deterministic generators does not eliminate the determinism but can increase complexity and the period of the output sequence.
– Techniques such as running one generator “randomly” stopping another and then selecting outputs can improve statistical properties and make patterns harder to discern, but they do not produce true unpredictability.
– True randomness requires a source of entropy fundamentally beyond deterministic algorithms—typically physical or quantum processes.
– Combining PRNGs may yield a longer period and pass more statistical randomness tests, but the sequence remains reproducible given the initial seeds and algorithmic states, making it pseudorandom by definition.

So, while your creative method could improve the quality of random-like output from a programming environment with limited tools, the final output was still not truly random but a complex pseudorandom sequence.

AI prompt 04 (follow-up):

does it means that physical world is separate from underlying, eternal mathematical properties of existence, from the “Platonic Space”?

AI response 04:

The idea that the physical world might be separate from an underlying eternal realm of mathematical truths aligns closely with the philosophical position known as **Mathematical Platonism**. According to mathematical platonism:

– Abstract mathematical objects (like numbers, sets, and mathematical truths) exist independently of space, time, and physical reality. They inhabit a non-physical realm often called the “Platonic realm” or “Platonic space”.
– These mathematical entities are timeless, changeless, and causally inert, meaning they do not physically interact with the world but are discovered rather than invented by humans.
– The physical universe, while described by mathematical laws, is considered distinct from this realm of mathematical objects. The universe might be seen as “instantiating” or reflecting these eternal mathematical structures, but not identical to them.
– This philosophical view implies a dual aspect of existence: the tangible, spatiotemporal physical world and the intangible, eternal mathematical realm.
– Plato himself conceived physical reality as a shadow or copy of perfect, eternal forms, including mathematical forms.

In sum, contemporary mathematical platonism supports the notion that physical reality is distinct from but mathematically structured according to an eternal, abstract “Platonic Space” of mathematical objects, which exists independently of the physical world as we experience it.

This does not conflict with the fact that physics uses mathematics extensively; rather, it highlights a metaphysical distinction between the realm of mathematical truths and the empirical universe.

/…/

[ here is missing a part of the discussion, including Gödel’s incompleteness theorems, mathematical formulas that generate chaotic systems like the butterfly effect, etc. — unfortunately i failed to copy some of the responses by AI, thus for consistency i better don’t include the entire discussion, continuing to the end of the conversation only ]

/…/

AI prompt 11 (follow-up):

are the physical random events at quantum level proven to be truly random in principle, or there can be undiscovered mathematical truths governing the physically manifest behavior, at the scale we are not able to reach with our current technology?

AI response 11:

The current scientific consensus, supported by strong experimental evidence, is that quantum events exhibit **true randomness in principle**. This means the outcomes of individual quantum measurements are fundamentally undecidable and incomputable by any algorithm or formula, making them genuinely unpredictable.

Key points from recent research include:

– Quantum randomness is proven to be **incomputable**, meaning no algorithm can reproduce the exact sequence of quantum measurement outcomes, even in principle.
– Experiments using quantum contextuality and violations of fundamental inequalities (such as the Kochen-Specker theorem) certify that the randomness is intrinsic and not due to hidden classical noise or undetected variables.
– These results suggest quantum randomness is not just a lack of current knowledge or technological limitation but a fundamental feature of nature.
– While some interpretations or speculative theories propose “hidden variables” or underlying deterministic laws, no experimental evidence has substantiated this, and standard quantum mechanics treats this randomness as irreducible.

In summary, though science remains open to new discoveries, current theoretical and experimental foundations strongly support that quantum randomness is truly random in principle, rather than being governed by undiscovered deterministic mathematical truths lying beyond present technological reach.

AI prompt 12 (follow-up):

you seem to confuse ‘consensus’ with a ‘proof’. are you saying there is nothing more to discover in the specific field?

AI response 12:

Scientific consensus about quantum randomness is based on extensive theoretical foundations and experimental verification, but it is not an absolute mathematical proof. There remain open philosophical and scientific questions:

– Some interpretations of quantum mechanics suggest underlying deterministic processes, though these are not yet supported by experimental evidence and often require nonlocal “hidden variables” which clash with current understanding of relativity.
– The consensus that quantum events are intrinsically random hinges on current empirical results and the framework of standard quantum theory, which has repeatedly passed experimental tests. However, science remains open to new discoveries and conceptual breakthroughs that could revise this understanding.
– In this sense, consensus reflects the best current understanding but does not preclude the possibility of future discoveries revealing deeper mathematical laws or structures beneath apparent randomness.
– This openness is a normal part of scientific rigor. No theory is ever conclusively final; rather, it is accepted until better explanatory models arise.

So while quantum randomness is currently viewed as truly fundamental and irreducible, it cannot be said with absolute finality that no more fundamental deterministic or mathematical explanation might ever be discovered. The scientific method embraces such possibilities pending evidence.

Thus, there is indeed still room for discovery in this field, but any new claims would require rigorous proof or empirical validation to supplant the current well-established probabilistic framework of quantum mechanics.

AI prompt 13 (follow-up):

thanks

AI response 13:

Scientific research has provided strong experimental evidence that quantum randomness is intrinsic and not just apparent due to lack of knowledge or technological limits. Quantum experiments, especially Bell tests and studies of entangled particles, demonstrate outcomes that are truly unpredictable and violate classical deterministic assumptions. Notably, recent advances have produced publicly accessible quantum random number generators certified by these principles, showing randomness that is traceable and verifiable by quantum physics itself.

However, the question of whether absolute deterministic “hidden variables” or undiscovered mathematical structures govern these events remains philosophically open. Some interpretations, like Bohmian mechanics, propose nonlocal hidden variables, but no experimental proof supports them, and they require strange physics beyond current mainstream views.

In summary:

– Quantum randomness is currently regarded as truly fundamental and irreducible by the majority of the scientific community and is experimentally supported as such.
– This does not equate to a mathematical proof that no deeper deterministic description is possible; rather, it reflects the best current evidence and understanding.
– Science remains open to future discoveries that might revise this view but any such changes would require solid empirical verification.

Thus, quantum randomness is both conceptually and practically treated as truly random today while acknowledging the inherent openness of scientific inquiry.

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