# The Utterly Prosaic Connection between Physics and Mathematics

## Abstract

**:**

## 1. Background

## 2. Quantum Conundrums

#### 2.1. Quantum Pedagogy

#### 2.2. Quantum Foundations

“...the physicists of [Einstein’s] own world had turned away from his effort and its failure, pursuing the magnificent incoherencies of quantum theory, with its high technological yields, ... to arrive at a dead end, a catastrophic failure of the imagination.”— Shevek circa 2500 CE

#### 2.3. Quantum Field Theory

“In the thirties, under the demoralizing influence of quantum-theoretic perturbation theory, the mathematics required of a theoretical physicist was reduced to a rudimentary knowledge of the Latin and Greek alphabets.”— Res Jost circa 1964

“I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce.”— Freeman Dyson 1972

- The fact that the Feynman expansion cannot possibly converge, (it is at best an asymptotic expansion even after you renormalize to effectively make each individual Feynman diagram finite), is probably just an annoyance...
- Haag’s theorem is still an obstruction to constructing a fully relativistic interaction picture, rather completely undermining standard textbook presentations of how to derive the Feynman diagram expansion. This may have been fixed (or rather, side-stepped) as of 2017 [7].
- As of 2018, not one single non-trivial interacting relativistic quantum field theory has rigorously been established to exist in 3 + 1 dimensions; though rigorous constructions are available in 2 + 1 and 1 + 1 dimensions. (The technical difference seems to be that there are interesting super-renormalizable quantum field theories in (2 + 1) and (1 + 1) dimensions; but that the rigorous techniques used to establish these results to not quite extent to the physically interesting, but merely renormalizable, quantum field theories in 3 + 1 dimensions.)

## 3. Usability Versus Precision

## 4. Instrumentalism Versus Realism

## 5. Mathematical Universe Hypothesis

“God created the integers; all else is the work of man.”—Leopold Kronecker

“Don’t let me catch anyone talking about the universe in my department.”—Ernest Rutherford

## 6. Discussion

“And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of Nature, and they ought to be least neglected in Geometry.”— Johannes Kepler

## Funding

## Conflicts of Interest

## Appendix A Technical End-Notes

#### Appendix A.1. Classical Uncertainty

#### Appendix A.2. Classical Barrier Penetration

## References

- Wigner, E. The unreasonable effectiveness of mathematics in the natural sciences. Commun. Pure Appl. Math.
**1960**, 13, 1–14. [Google Scholar] [CrossRef] - Visser, M. Which number system is ‘best’ for describing empirical reality? Math. Phys. arXiv:1212.6274.
- Wilber, K. Quantum Questions: Mystical Writings of the World’s Great Physicists; Shambhala Publications: Boston, MA, USA, 2001; ISBN-10 1570627681, ISBN-13 978-1570627682. [Google Scholar]
- Green, M. Quantum Physics and Ultimate Reality: MysticalWritings of Great Physicists; Amazon Kindle Publishing: Seattle, WA, USA, 2014. [Google Scholar]
- Stenger, V. Quantum Quackery. Sceptical Inquirer 2.1 (January/February 1997). Available online: http://www.csicop.org/si/show/quantum_quackery/ (accessed on 18 September 2018).
- Le Guin, U. This is one of the few novels to make a research physicist the central character, and do so in a subtle and convincing manner. In The Dispossessed; HarperCollins Publishers: New York, NY, USA, 1974. [Google Scholar]
- Seidewitz, E. Parameterized quantum field theory without Haag’s theorem. Found. Phys.
**2017**, 45, 35–374. [Google Scholar] [CrossRef] - Tegmark, M. The Mathematical Universe. Found. Phys.
**2008**, arXiv:0704.064638, 101. [Google Scholar] - Tegmark, M. Shut up and calculate. Pop. Phys.
**2007**, arXiv:0709.4024. [Google Scholar] - Visser, M. Lorentzian Wormholes: From Einstein to Hawking; AIP Press: New York, NY, USA, 1995. [Google Scholar]

1. | An example where physics/engineering led mathematics (by about two decades) was Fourier’s study of heat transfer, which led to the introduction of Fourier series in 1807, followed by rigourous mathematical work by Dirichlet in 1829 to confirm existence and convergence of these series under suitable conditions, followed by wholesale adoption of these techniques by the physics/engineering/applied mathematics communities. Similarly, a six-decade example was initiated in the late 1890’s with the Heaviside calculus, which led to Dirac’s delta functions, which were then not made fully rigorous until mathematicians developed Schwartz distribution theory in the 1950s. In counterpoint, an example where mathematics led physics was with the development of Riemann’s differential geometry beginning in the 1850s, a mathematical structure that was then adapted by Einstein in 1915 to set up his general theory of relativity, again a six-decade timescale. |

2. | While Planck basically introduced the notion that electromagnetic radiation could interact with matter only in “little lumps”, eventually called “quanta”, it was Einstein and De Broglie who made much more quantitative and precise statements that the energy and momentum of the “little lumps” of electromagnetic radiation were proportional to the frequency and the reciprocal of the wavelength—this then immediately leads to the quantitative version of the Heisenberg uncertainty relation. For some technical notes addressing this point see Appendix A.1. |

3. | Without some notion of collapse to define definite outcomes/events, there is simply no basis for any notion of memory or history. Without memory/history there is no notion of cause and effect. Without being able to separate cause and effect, I have no idea how to even begin to define causality. |

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**MDPI and ACS Style**

Visser, M.
The Utterly Prosaic Connection between Physics and Mathematics. *Philosophies* **2018**, *3*, 25.
https://doi.org/10.3390/philosophies3040025

**AMA Style**

Visser M.
The Utterly Prosaic Connection between Physics and Mathematics. *Philosophies*. 2018; 3(4):25.
https://doi.org/10.3390/philosophies3040025

**Chicago/Turabian Style**

Visser, Matt.
2018. "The Utterly Prosaic Connection between Physics and Mathematics" *Philosophies* 3, no. 4: 25.
https://doi.org/10.3390/philosophies3040025