Abstract
Mathematics is usually regarded as a kind of language. The essential behavior of physical phenomena can be expressed by mathematical laws, providing descriptions and predictions. In the present essay I argue that, although mathematics can be seen, in a first approach, as a language, it goes beyond this concept. I conjecture that mathematics presents two extreme features, denoted here by irreducibility and insaturation, representing delimiters for self-referentiality. These features are then related to physical laws by realizing that nature is a self-referential system obeying bounds similar to those respected by mathematics. Self-referential systems can only be autonomous entities by a kind of metabolism that provides and sustains such an autonomy. A rational mind, able of consciousness, is a manifestation of the self-referentiality of the Universe. Hence mathematics is here proposed to go beyond language by actually representing the most fundamental existence condition for self-referentiality. This idea is synthesized in the form of a principle, namely, that mathematics is the ultimate tactics of self-referential systems to mimic themselves. That is, well beyond an effective language to express the physical world, mathematics uncovers a deep manifestation of the autonomous nature of the Universe, wherein the human brain is but an instance.
To the question which is older, day or night, he [Thales of Miletus] replied: — Night is the older by one day.
Thales of Miletus [624 BC–546 BC]
Either mathematics is too big for the human mind or the human mind is more than a machine.
Kurt Gödel
This essay received the 4th. Prize in the 2015 FQXi essay contest: “Trick or Truth: the Mysterious Connection Between Physics and Mathematics”.
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Notes
- 1.
For occurrences of the Golden Ratio in the natural world, see, e.g., Ref. [6]. For a beautiful example of two different partial differential equations (PDEs) describing the same physical phenomenon, see Ref. [7]. A limiting procedure that suggests an explanation for the wide applicability and universality of some integrable PDEs can be found in [8].
- 2.
Although very useful, mathematics would be ultimately limited to transform “emptiness into emptiness” (see commentary by Frèchet in The Mathematical Thought, part IV of Ref. [3]).
- 3.
I am tempted to assert that such kinds of proofs are actually impossible.
- 4.
See, e.g.: http://en.wikipedia.org/wiki/Impredicativity and http://en.wikipedia.org/wiki/Self-reference, respectively.
- 5.
That statement is assumed valid in terms of circular quantifications only. According to Wikipedia, “The vicious circle principle is a principle that was endorsed by many predicativist mathematicians in the early 20th century to prevent contradictions. The principle states that no object or property may be introduced by a definition that depends on that object or property itself.” It was later realized that circularity in terms of quantification does not lead to paradoxes, as such definitions do not “create sets or properties or objects, but rather just give one way of picking out the already existing entity from the collection of which it is a part” (c.f. http://en.wikipedia.org/wiki/Vicious_circle_principle).
- 6.
It is just like “looking down the wholeness of oneself from a great height”, but, then, with vertigo, one cannot actually look.
- 7.
Here I do not desire to engage in a discussion of various philosophical positions on the nature of time or determinism. Notice that the conjectures do not strictly depend on those.
- 8.
Sadly, he had an unfortunate end, being executed by the Gestapo at the end of WWII. Details can be found in Ref. [3], or on Wikipedia, http://en.wikipedia.org/wiki/Jean_Cavailles.
- 9.
Indeed, it is quite difficult to characterize mathematics, at the level of rigor that it attains as its natural condition, by some different means. For example, according to Cavaillès [3]: (i) mathematics is not exactly a part of logic, since some mathematical notions may be combinatorial in nature, or based on other notions irreducible to purely logical operations—yet, those are still mathematical anyway. And (ii) mathematics cannot be entirely characterized as a hypothetical-deductive system, yet alternative/complementary notions for that purpose are still mathematical anyway.
- 10.
- 11.
We allow for the existence of infinitely countable predicatives within the system, as particular cases of impredicatives that are self-referring to a null set.
- 12.
Far from bringing ideological assumptions to the matter, compare with Marx’s quote: “Labour (...) is an eternal natural necessity which mediates the metabolism between man and nature, and therefore human life itself” [11]. This made me think of a parallel reasoning for the connection of mathematics and nature.
- 13.
Indeed, one may argue that a sufficiently advanced technology must be indistinguishable from nature. That would be an expected trend, according to the arguments of the present essay. http://www.universetoday.com/93449/do-alien-civilizations-inevitably-go-green/.
- 14.
References
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Acknowledgments
The author thanks Fabiano L. de Sousa for useful suggestions.
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Dantas, C.C. (2016). The Ultimate Tactics of Self-referential Systems. In: Aguirre, A., Foster, B., Merali, Z. (eds) Trick or Truth?. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-27495-9_17
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