# Contemporary Natural Philosophy and Contemporary Idola Mentis

## Abstract

**:**

Dedicated to Gordana Dodig-Crnkovic |

who proposed the idea of the Contemporary Natural Philosophy Project |

## 1. Introduction

## 2. Contemporary Natural Philosophy

#### 2.1. Motivations for Contemporary Natural Philosophy

#### 2.2. Philosophical Framework of Contemporary Natural Philosophy

The traditional sciences have always had trouble with ambiguity. To overcome this barrier, ‘science’ has imposed ‘enabling constraints’—hidden assumptions which are given the status of ceteris paribus. Such assumptions allow ambiguity to be bracketed away at the expense of transparency. These enabling constraints take the form of uncritically examined presuppositions, which we refer to throughout the article as ‘uceps.’ […] Second order science reveals hidden issues, problems and assumptions which all too often escape the attention of the practicing scientist (but which can also get in the way of the acceptance of a scientific claim) [17].

The tenets of realist liberal naturalism are: (i) A liberalized ontological tenet, according to which some real and non-supernatural entities exist that are irreducible to the entities that are part of the coverage domain of a natural science-based ontology; (ii) A liberalized epistemological tenet, according to which some legitimate forms of understanding (say, a priori reasoning or introspection) are neither reducible to scientific understanding nor incompatible with it; (iii) A liberalized semantic tenet, according to which there are linguistic terms that refer to real non-supernatural entities that do not form part of the coverage domain of natural science and are not reducible to those entities which do; (iv) A liberalized metaphilosophical tenet, according to which there are issues in dealing with which philosophy is not continuous with science as to its content, method and purpose [22].

## 3. From Baconian Idola Mentis to Contemporary Idola Mentis

I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction [29].

## 4. The Idols of the Number

#### 4.1. What Do We Know When We Know the Number?

Modern mathematics is the formal study of structures that can be defined in a purely abstract way. Think of mathematical symbols as mere labels without intrinsic meaning. It doesn’t matter whether you write ‘two plus two equals four’, ‘2 + 2 = 4′ or ‘dos mas dos igual a cuatro’. The notation used to denote the entities and the relations is irrelevant; the only properties of integers are those embodied by the relations between them. That is, we don’t invent mathematical structures—we discover them, and invent only the notation for describing them. So here is the crux of my argument. If you believe in an external reality independent of humans, then you must also believe in what I call the mathematical universe hypothesis: that our physical reality is a mathematical structure. In other words, we all live in a gigantic mathematical object […] [32],

#### 4.2. Numbers and Their Structures

_{+}

^{−1}= −a with the traditional notation a

^{−1}for the inverse coming from the fact that the inverse for a non-0 real number a is its reciprocal 1/a = a

^{−1}. The nullary operation does not require any choice of arguments as its value is independent from arguments and consists in the selection of some constant element, for instance, the choice of 0 or choice of 1 which both have special roles of the neutral element as defined below. Notice that to define an operation on a set requires that for all arguments there is an outcome of the operation.

- ⮚
- (no name as it is the universal condition for operation) ∀a,b∈S∃c∈S: ab = c
- ⮚
- associativity can be written simply ∀a,b∈S∃c∈S: ab = c ∀a,b,c∈S: (ab)c = a(bc).

^{−1}for an element a.

^{−1}satisfying the condition: aa

^{−1}= a

^{−1}a = e is called an inverse of a.

_{+}

^{−1}>. This group is called a commutative group, because for all real numbers: a + b = b + a.

_{●}

^{−1}> with respect to multiplication ● on the subset R* = R\{0} is commutative. Obviously, the subset R* of R is closed with respect to multiplication, i.e., ∀a,b∈R*: ab∈R* and ∀a∈R*: aa

^{−1}= a

^{−1}a=1 when a

^{−1}= 1/a.

_{+}

^{−1}, ●, 1, a → a

_{●}

^{−1}> where <R, +, 0, a → a

_{+}

^{−1}> is its additive commutative group and <R*, ●, 1, a → a

_{●}

^{−1}> is its commutative multiplicative group. This type of an algebraic structure is called a field if ∀a,b,c∈S: a(b + c) = ab + ac, i.e., multiplication is distributed over addition.

_{+}

^{−1}, ●, 1, a → a

_{●}

^{−1}> (if no confusion is likely, we will write shorter: <K, +, 0, ●, 1 >) defined not necessarily on real numbers but on a set K, where <K, +, 0, a → a

_{+}

^{−1}> is a commutative group (we say the additive group of the field) and K* is a commutative group <K*, ●,1, a → a

_{●}

^{−1}> where K* = K\{0} (we say the multiplicative group of the field). We combine these two groups with the requirement that multiplication is distributed over addition: ∀a,b,c∈K: a(b + c) = ab + bc.

_{+}

^{-1}, ●,1, a → a

_{●}

^{-1}> (shortly written <K, +, 0, ●, 1 >) can be found in many disciplines of mathematics and in many applications. The elements of a field K are what we call numbers or scalars, but this status is dependent not on individual elements but on the membership in the algebraic structure. It was already mentioned above that for the Ancient Greeks, numbers were elements of the field <Q, +, 0, ●, 1> and it took more than two millennia to extend this field to the clearly defined field <R, +, 0, ●, 1>. For us, it is important that there are several important examples of infinite fields between the field of rational numbers <Q, +, 0, ●, 1> and the field of real numbers <R, +, 0, ●, 1>, i.e., these fields form a chain of consecutive extensions or consecutive subfields of the field of rational numbers which in turn are subfields of real numbers.

_{+}

^{−1}, ●, 1, a → a

_{●}

^{−1}> (in short <K, +, ●>) in the chain considered above are defined on some proper subsets K of real numbers (K ⊆ R and K ≠ R) starting from the field of rational numbers Q. We can easily, and in the full agreement with our intuition, construct rational numbers forming the set Q from the integers in Z which in turn can be easily derived from the natural numbers in N. Of course, neither the set of natural numbers nor set of integers has the structure of a field with operations +, ● as they lack multiplicative inverses. The field of rational numbers Q is the smallest field including all natural numbers.

^{3}√2 which are roots of polynomials with rational coefficients (real algebraic field) which are not constructible. For instance,

^{3}√2 is a solution of the equation x

^{3}= 2. Thus, when the elements of reality started to be considered in terms of equations, it was necessary to search for further extension. The next larger field is the field of computable real numbers which can be results of the work of a Turing Machine, i.e., the work of any computer. It is countable, so still much smaller than the uncountable field of the real numbers <R, +, ●>. The majority of real numbers are not computable. Even worse, the majority of real numbers are not definable. Between the field of computable numbers and the field of the real numbers, there is a countable field of the definable numbers. These are numbers which can be identified by a description in terms of logic and set theory. The uncountable majority of real numbers are not definable. There is no way we can identify non-definable numbers. They do not have any properties expressible in mathematical language which we could use to distinguish them.

#### 4.3. Unreasonable Misunderstandings of Mathematics

## 5. Idols of the Common Sense

#### 5.1. Beware of What Escapes Awareness

#### 5.2. Definition of the Definition

_{i}log p

_{i}(the constant K (omitted in the formula, m.j.s) merely amounts to a choice of a unit of measure) play a central role in information theory as measures of information, choice and uncertainty” [54] (p. 20). There is not much more directly about information in this historical paper, yet it is considered that Shannon defined here “information”. It is clear that the two idols, of the Number and of the Common Sense, are responsible for this opinion. The former prompts people to believe that something expressed as a number giving value to some magnitude must be an entity. The latter idol just obscures the meaning of the definition as a concept.

## 6. The Idols of the Elephant

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Dodig-Crnkovic, G.; Schroeder, M.J. (Eds.) Contemporary Natural Philosophy and Philosophies, Part 1; MDPI: Basel, Switzerland, 2019; Available online: https://www.mdpi.com/books/pdfview/book/1331 (accessed on 30 June 2020).
- Dodig-Crnkovic, G.; Schroeder, M.J. Contemporary natural philosophy and philosophies. Philosophies
**2018**, 3, 42. [Google Scholar] [CrossRef][Green Version] - Fallacies. Internet Encyclopedia of Philosophy; American Library Association: Washington, DC, USA, 1995; Available online: https://iep.utm.edu/fallacy/) (accessed on 16 August 2020).
- Snow, C.P. The Two Cultures; Cambridge University Press: London, UK, 1959. [Google Scholar]
- Conant, J.B. General Education in a Free Society: Report of the Harvard Committee; Harvard University Press: Cambridge, MA, USA, 1946. [Google Scholar]
- Wilson, E.O. Consilience: The Unity of Knowledge; Vintage Books: New York, NY, USA, 1999. [Google Scholar]
- Wilson, E.O. The Meaning of Human Existence; Liveright Publishing: New York, NY, USA, 2014. [Google Scholar]
- Bateson, G. Mind and Nature: A Necessary Unity; E.P. Dutton: New York, NY, USA, 1979. [Google Scholar]
- Butler, D. Theses spark twin dilemma for physicists. Nature
**2002**, 420, 5. [Google Scholar] [CrossRef] [PubMed] - Wilczek, F. Physics in 100 Years. Available online: http://frankwilczek.com/2015/ physicsOneHundredYears03.pdf (accessed on 17 August 2020).
- Simeonov, P.L.; Smith, L.S.; Ehresmann, A.C. Stepping Beyond the Newtonian Paradigm in Biology: Towards an Integrable Model of Life–accelerating discovery in the biological foundations of science, INBIOSA White Paper. In Integral Biomathics: Tracing the Road to Reality; Simeonov, P.L., Smith, L.S., Ehresmann, A.C., Eds.; Springer: Berlin/Heidelberg, Germany, 2012; pp. 319–418. [Google Scholar]
- Capra, F. The Tao of Physics; Shambhala Publications: Boulder, CO, USA, 1975. [Google Scholar]
- Laplane, L.; Mantovani, P.; Adolphs, R.; Chang, H.; Mantovani, A.; McFall-Ngai, M.; Rovelli, C.; Sober, E.; Pradeu, T. Why science needs philosophy. Proc. Natl. Acad. Sci. USA
**2019**, 116, 3948–3952. [Google Scholar] [CrossRef][Green Version] - Von Foerster, H. (Ed.) Cybernetics of cybernetics: Or, the Control of Control and the Communication of Communication, 2nd ed.; Future Systems: Minneapolis, MN, USA, 1995. [Google Scholar]
- Umpleby, S.A. Second-order science: Logic, strategies, methods. Construct. Found.
**2014**, 10, 16–23. [Google Scholar] - Müller, K.H. Second-order Science: The Revolution of Scientific Structures; Echoraum: Wien, Austria, 2016. [Google Scholar]
- Lissack, M. Second order science: Examining hidden presuppositions in the practice of science. Found. Sci.
**2017**, 22, 557–573. [Google Scholar] [CrossRef] - Quine, W.V.O. Mr. Strawson on logical theory. Mind
**1953**, 62, 433–451. [Google Scholar] [CrossRef] - Laudan, L. Science and Relativism: Some Key Controversies in the Philosophy of Science; The University of Chicago Press: Chicago, IL, USA, 1990. [Google Scholar]
- Niiniluoto, I. Critical Scientific Realism; Clarendon Press: Oxford, UK, 1999. [Google Scholar]
- Dummett, M. Realism. Synthese
**1982**, 52, 55–112. [Google Scholar] [CrossRef] - De Caro, M. Naturalism and Realism. In A Companion to Naturalism; do Carmo, J.S., Ed.; Dissertatio’s Series of Philosophy; University of Pelotas: Pelotas, Brasil, 2016; pp. 182–195. [Google Scholar]
- Schroeder, M.J. Invariance as a tool for ontology of information. Information
**2016**, 7, 11. Available online: https://www.mdpi.com/2078-2489/7/1/11 (accessed on 15 June 2020). [CrossRef][Green Version] - Lyotard, J.-F. Theory and history of literature. In The Postmodern Condition: A Report on Knowledge; University of Minnesota Press: Minneapolis, MN, USA, 1984; Volume 10. [Google Scholar]
- Anderson, P. The Origins of Postmodernity; Verso: London, UK, 1998; pp. 24–27. [Google Scholar]
- Bacon, F. Novum Organum or True Suggestions for the Interpretation of Nature; Devey, J., Ed.; P.F. Colier & Son: New York, NY, USA, 1902; Available online: http://www.gutenberg.org/files/45988/45988-h/ 45988-h.htm (accessed on 7 June 2020).
- Bacon, R. Opus Majus of Roger Bacon: Part I; Kessinger Publishing: Whitefish, MT, USA, 2002. [Google Scholar]
- Whorf, B.L. Language, Thought, and Reality; Caroll, J.B., Ed.; MIT Press: Cambridge, MA, USA, 1956. [Google Scholar]
- Newton, I. Philosophiae Naturalis Principia Mathematica, General Scholium, 3rd ed.; University of California Press: Berkeley, CA, USA, 1726/1999; ISBN 0-520-08817-4. [Google Scholar]
- Losee, J. A Historical Introduction to the Philosophy of Science; Oxford Univ. Press: London, UK, 1972. [Google Scholar]
- Adams, D. The Hitchhiker’s Guide to the Galaxy; Pan Macmillan Books: London, UK, 1979. [Google Scholar]
- Tegmark, M. Shut up and calculate. arXiv
**2007**, arXiv:physics/0709.4024v1. [Google Scholar] - Schroeder, M.J. Crisis in science: In search for new theoretical foundations. Progress Biophys. Mol. Biol.
**2013**, 113, 25–32. [Google Scholar] [CrossRef] [PubMed] - Schroeder, M.J. Structures and their cryptomorphic manifestations: Searching for inquiry tools. In Algebraic Systems, Logic, Language and Related Areas in Computer Science, RIMS Kokyuroku; Adachi, T., Ed.; Kyoto Research Institute for Mathematical Sciences, Kyoto University: Kyoto, Japan, 2019; p. 10. [Google Scholar]
- Schroeder, M.J. Equivalence, (crypto) morphism and other theoretical tools for the study of information. Proceedings
**2020**, 47, 12. [Google Scholar] [CrossRef] - Schroeder, M.J. Intelligent computing: Oxymoron? Proceedings
**2020**, 47, 31. [Google Scholar] [CrossRef] - Schroeder, M.J. Logico-algebraic structures for information integration in the brain. In Algebras, Languages, Computation and Their Applications, RIMS Kokyuroku; Shoji, K., Ed.; Research Institute for Mathematical Sciences, Kyoto University, Kyoto: Kyoto, Japan, 2007; Volume 1562, pp. 61–72. Available online: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/1562.html (accessed on 20 June 2020).
- Wigner, E. The unreasonable effectiveness of mathematics in the natural sciences. In Communications in Pure and Applied Mathematics; New York University: New York, NY, USA, 1960; Volume 13. [Google Scholar]
- Weyl, H. Symmetry; Princeton University Press: Princeton, NJ, USA, 1952. [Google Scholar]
- Anderson, P.W. More is different. Science
**1972**, 177, 393–396. [Google Scholar] [CrossRef] [PubMed][Green Version] - Schroeder, M.J. Concept of information as a bridge between mind and brain. Information
**2011**, 2, 478–509. Available online: https://www.mdpi.com/2078-2489/2/3/478/ (accessed on 1 June 2020). [CrossRef][Green Version] - Schroeder, M.J. Towards cyber-phenomenology: Aesthetics and natural computing in multi-level information systems. In Recent Advances in Natural Computing; Ser. Mathematics for Industry 9; Suzuki, Y., Hagiya, M., Eds.; Springer: Tokyo, Japan, 2015; pp. 69–86. [Google Scholar]
- Tversky, A.; Kahneman, D. Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. Psychol. Rev.
**1983**, 90, 293–315. [Google Scholar] [CrossRef] - Suppes, P. Introduction to Logic; Van Nostrand: Princeton, NJ, USA, 1957. [Google Scholar]
- Simon, H.A. The axiomatization of physical theories. Philos. Sci.
**1970**, 37, 16–26. [Google Scholar] [CrossRef] - Tarski, A. Some methodological investigations on the definability of concepts. In Logic Semantics, and Metamathematics: Papers from 1923 to 1938; The Clarendon Press: Oxford, UK, 1956. [Google Scholar]
- Carroll, L. Alice’s Adventures in Wonderland; Pan Macmillan Books: London, UK, 1865. [Google Scholar]
- Kroeber, A.L. Culture: A critical review of concepts and definitions. In Papers of the Peabody Museum of American Archaeology and Ethnology; Harvard University, Peabody Museum of American Archaeology: Andover, MA, USA, 1952; Volume 42. [Google Scholar]
- Lovejoy, A.O. Nature as aesthetic norm. Mod. Lang. Notes
**1927**, 7, 444–450. [Google Scholar] [CrossRef] - Lovejoy, A.O. Some meanings of ‘nature’. In A Documentary History of Primitivism and Related Ideas; Lovejoy, A.O., Boas, G., Eds.; Johns Hopkins Press: Baltimore, MD, USA, 1935; pp. 447–456. [Google Scholar]
- Williams, R. Ideas of nature. In Problems in Materialism and Culture: Selected Essays; Williams, R., Ed.; Verso: London, UK, 1980; pp. 67–85. [Google Scholar]
- Williams, R. Keywords: A Vocabulary of Culture and Society, rev. ed.; Bennet, T., Grossberg, L., Morris, M., Eds.; Blackwell: Malden, MA, USA, 2005; pp. 235–238. [Google Scholar]
- Ogden, C.K.; Richards, I.A. The Meaning of Meaning: A Study of the Influence of Language Upon Thought and of the Science of Symbolism; Harcourt Brace Jovanovich: San Diego, CA, USA, 1989. [Google Scholar]
- Shannon, E.C.; Weaver, W. The Mathematical Theory of Communication; University of Illinois Press: Urbana, IL, USA, 1949. [Google Scholar]
- Kosso, P. Science and Objectivity. J. Philos.
**1989**, 86, 245–257. [Google Scholar] [CrossRef] - Schroeder, M.J. Hierarchic information systems in a search for methods to transcend limitations of complexity. Philosophies
**2016**, 1, 1–14. Available online: https://www.mdpi.com/2409-9287/1/1/1 (accessed on 15 June 2020). [CrossRef][Green Version] - Schroeder, M.J. Exploring meta-symmetry for configurations in closure spaces. In Developments of Language, Logic, Algebraic System and Computer Science, RIMS Kokyuroku; Horiuchi, K., Ed.; Kyoto Research Institute for Mathematical Sciences, Kyoto University: Kyoto, Japan, 2017; pp. 35–42. Available online: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2051-07.pdf (accessed on 1 June 2020).
- Worrall, J. Structural Realism: The best of both worlds? Dialectica
**1989**, 43, 99–124. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Schroeder, M.J.
Contemporary Natural Philosophy and Contemporary *Idola Mentis*. *Philosophies* **2020**, *5*, 19.
https://doi.org/10.3390/philosophies5030019

**AMA Style**

Schroeder MJ.
Contemporary Natural Philosophy and Contemporary *Idola Mentis*. *Philosophies*. 2020; 5(3):19.
https://doi.org/10.3390/philosophies5030019

**Chicago/Turabian Style**

Schroeder, Marcin J.
2020. "Contemporary Natural Philosophy and Contemporary *Idola Mentis*" *Philosophies* 5, no. 3: 19.
https://doi.org/10.3390/philosophies5030019