Dark energy as a manifestation of nontrivial arithmetic

Marek CzachorPolitechnika Gdańska

(KCIK 2016)

Flat Friedman-Robertson-Walker universea(t)

t

GR without DE

Observable expansion

Geometric Kant problem

● Different alternative geometries are allowed by mathematics.

● So, why does the Universe obey those particular rules invented by Euclid of Alexandria?

Arithmetic Kant problem

● Different alternative arithmetics are allowed by mathematics.

● So, why does the Universe obey those particular rules invented by Diophantus of Alexandria?

Arithmetic Kant problem

● Different alternative arithmetics are allowed by mathematics.

● So, why does the Universe obey those particular rules invented by Diophantus of Alexandria?

Mark Burgin Non-Diophantine Arithmetics, or is it possible that 2+2 ≠ 4?Kiev, Ukrainian AIS, 1997

Arithmetic Kant problem: Example

Take and define

Addition/subtraction is modified but multiplication is unchanged

Arithmetic Kant problem: Example

Take and define

Addition/subtraction is modified but multiplication is unchanged

Non-Diophantine

Diophantine

Then all the rules are preserved

supplemented by subtraction and division

associative

commutative

distributive

unit element

zero

etc.

Now change notation so that

Now change notation so that

So, which arithmetic is Diophantine?

Which arithmetic is “more natural”, “simpler”, “more physical”, or “given a priori” (Kant)?

Which arithmetic is “more natural”, “simpler”, “more physical”, or “given a priori” (Kant)?

This?

Which arithmetic is “more natural”, “simpler”, “more physical”, or “given a priori” (Kant)?

Or this?

Derivative

Or perhaps it's all irrelevant?

If all mathematical operations performed by “us” and by the Nature involve the same arithmetic and calculus, there may be no means of detecting a nontrivial f

Or perhaps it's all irrelevant?

If all mathematical operations performed by “us” and by the Nature involve the same arithmetic and calculus, there may be no means of detecting a nontrivial f

But what if there is a miss-match between a natural arithmetic and the one we employ in mathematical modeling?

Or perhaps it's all irrelevant?

If all mathematical operations performed by “us” and by the Nature involve the same arithmetic and calculus, there may be no means of detecting a nontrivial f

But what if there is a miss-match between a natural arithmetic and the one we employ in mathematical modeling?

Are there any examples of such a miss-match in any natural science?

Or perhaps it's all irrelevant?

If all mathematical operations performed by “us” and by the Nature involve the same arithmetic and calculus, there may be no means of detecting a nontrivial f

But what if there is a miss-match between a natural arithmetic and the one we employ in mathematical modeling?

Are there any examples of such a miss-match in any natural science?

Yes, and we experience it in our everyday life!

Weber-Fechner lawE. H. Weber, in Handlewörterbuch der Physiologie, iii. (1846)G. T. Fechner, Elemente der Psychophysik (1860)

Weber-Fechner lawE. H. Weber, in Handlewörterbuch der Physiologie, iii. (1846)G. T. Fechner, Elemente der Psychophysik (1860)

WeightSoundVisionTaste...

EconomicsPolitics...

Confirmed in psychological studies of subjective perception of change and in objective neurophysiologicalmeasurements

Weber-Fechner problem in arithmetic formG. T. Fechner, Elemente der Psychophysik (1860)M. Czachor, Information processing and Fechner's problem as a choice of arithmetic, arXiv:1602.00587 [q-bio.NC] (Neurons and Cognition)

Find a generalized subtraction such that

is independent of x

The problem was formulated on 22 Oct 1850 and uniquely solved by Fechner,

This is why decibels correspond to a log scale

Weber-Fechner problem in arithmetic formG. T. Fechner, Elemente der Psychophysik (1860)M. Czachor, Information processing and Fechner's problem as a choice of arithmetic, arXiv:1602.00587 [q-bio.NC] (Neurons and Cognition)

Find a generalized subtraction such that

is independent of x

The problem was formulated on 22 Oct 1850 and uniquely solved by Fechner,

This is why decibels correspond to a log scale

All our sensations are perceived via a log channel

Weber-Fechner arithmetic

Neutral elements

Addition/subtraction

Negative numbers

Light-cone in Weber-Fechner-Minkowski space

Light-cone in Weber-Fechner-Minkowski space

Two arithmetics coexist in

● “Standard” Diophantine

● Non-Diophantine “Fechnerian”

Two space-times coexist in

● Locally Lorenzian manifold with Diophantine arithmetic/calculus and event horizons on the walls of

● Globally Minkowskian space with non-Diophantine arithmetic/calculus

The problem● Two types of arithmetic/calculus can coexist in the same space-time

● Which of them should be employed in mathematical formulation of equations of physics?

● Is it possible that we work with the wrong one?● If so, what are the observable consequences?

Minkowski space

Arithmetic defined by

Minkowski space

Arithmetic defined by

Apparent velocity of a static charge as seen from the center of the universe with L=1 (Maxwell equations with non-Diophantine calculus)

Solution of Friedman equation in flat FRW universe with L=20 (Einstein equations with non-Diophantine calculus, no dark energy, zero cosmological constant)

Flat Friedman-Robertson-Walker universea(t)

t

GR without DE

Observable expansion

The Final Big Question

What is the physical law that determines f ?

Arithmetic:The cleanest source of dark

energy

Marek CzachorPolitechnika Gdańska

(KCIK 2016)

Bibliography

● M. Czachor, Relativity of arithmetic as a fundamental symmetry of physics, Quantum Stud.: Math. Found. 3, 123-133 (2016), arXiv:1412.8583 [math-ph]

● D. Aerts, M. Czachor, M. Kuna, Crystallization of space: Space-time fractals from fractal arithmetic, Chaos, Solitons and Fractals 83, 201-211 (2016), arXiv:1506.00487 [gr-qc]