Skip to main content

Proposal for a Degree of Scientificity in Cosmology


In spite of successful tests, the standard cosmological model, the \(\varLambda\)CDM model, possesses the most problematic concept: the initial singularity, also known as the big bang. In this paper—by adopting the Kantian difference between to think of an object and to cognize an object—it is proposed a degree of scientificity using fuzzy sets. Thus, the notion of initial singularity will not be conceived of as a scientific issue because it does not belong to the fuzzy set of what is known. Indeed, the problematic concept of singularity is some sort of what Kant called the noumenon, but science, on the other hand, is constructed in the phenomenon. By applying the fuzzy degree of scientificity in cosmological models, one concludes that cosmologies with a contraction phase before the current expansion phase are potentially more scientific than the standard model. At the end of this article, it is shown that Kant’s first antinomy of pure reason indicates a limit to our cosmological models.

This is a preview of subscription content, access via your institution.


  1. \(\varLambda\) is the cosmological constant, and CDM means cold dark matter.

  2. Weinberg (2014, p. 1) defines typical freely falling observers as “those that move with the average velocity of typical galaxies in their respective neighborhoods.”

  3. For some authors, dark energy and dark matter are extra ingredients as well. According to Merritt (2017, p. 41), dark matter and dark energy “are auxiliary hypotheses that were invoked in response to observations that falsified the standard model.” The author considers dark energy and dark matter as conventionalist stratagems and non-falsifiable alternatives in cosmology.

  4. Problems like flatness, isotropy, and homogeneity in the observable universe, i.e., inflation may generate a flat, isotropic, and homogeneous universe. Moreover, the inflationary mechanism is a powerful way to provide quantum fluctuations in the early universe, which are considered as seeds for the structure formation.

  5. Popper’s philosophy within a Kantian or neo-Kantian tradition is found, for example, in Naraniecki (2010).

  6. An introduction to fuzzy sets and its applications is found in Zimmermann (1996, 2010).

  7. I adopt a common convention among Kant scholars. In passages of the Critique of pure reason, it is indicated the letter B, second edition, and the correspondent page number.

  8. An interesting discussion on the concepts of space and time in Kant, comparing them to the relativity’s point of view, is found in Dorato (2002). According to the author, it is possible to support a Kantian interpretation of space and time (or space–time) in the Einsteinian context.

  9. See van den Berg (2011), and Watkins and Stan (2014) for discussions on the notion of proper science in Kant.

  10. I suggest that the notion of field in modern physics plays the role of the notion of substance or substratum in Kant’s philosophy. According to the philosopher of Königsberg, “the substratum of everything real, i.e., everything that belongs to the existence of things, is substance, of which everything that belongs to existence can be thought only as a determination” (Kant 1998, B 225). A classical or quantum field cannot be “observed,” only its determinations or states (particles).

  11. In Timaeus, Plato describes the creation of the universe using a metaphor. The maker of the universe, Demiurge, acts through thought or ideal objects as patterns in order to construct the organized world, our cosmos, from chaos (see Plato 1931, 29a).

  12. See Anderson (1998) for an introduction to Nietzschean perspectivism and Neves (2019a) in which it is applied to physics. Babich (1999) presents studies on Nietzschean philosophy of science.

  13. See Pula (1992) for Nietzsche’s relation to Sapir–Whorf hypothesis in linguistics. The Sapir–Whorf hypothesis states that the structure of a language influences the speaker’s world view. In Twilight of the Idols, Nietzsche (2005, p. 170, “Reason” in philosophy, 5) emphasizes this point in a famous passage: “I am afraid that we have not got rid of God because we still have faith in grammar...”.

  14. Nietzsche (2002, p. 15, §14).

  15. One of the clearest passages in which Nietzsche states this point of view is in Beyond good and evil, aphorism 230: “(...) ‘spirit’ resembles a stomach more than anything.” This point is studied in Neves (2019a).

  16. Nietzsche (2003, p. 206), fragment 10 [202] of 1887.

  17. Nietzsche (2002, p. 22, §22).

  18. Nietzsche is among those who do not distinguish between thing in itself and noumenon. According to Nietzsche Source, the word noumenon does not appear in Nietzsche’ works. Thing in itself, on the contrary, is common in his books (see

  19. In this sense, Kant indicates a hypothetical intellectual intuition (Kant 1998, B 308).

  20. A similar conclusion is shown in Romero (2013), in which space–times singularities are interpreted as nonphysical realities and defective products of Einstein’s theory, since energy conditions are obeyed.

  21. Following Nietzsche, as the thing in itself is an absurd, then every object is given by the understanding’s properties and relations in statements.

  22. According to Leech (2017, p. 98), logical and real possibilities, in Kantian philosophy, mean that “something is logically possible just when the concept of it is non-contradictory, and something is really possible just when the concept of it is non-contradictory and consistent with the a priori constraints on experience arising from the forms of intuition and the categories.” In our science, the word logic does not present the same meaning compared to Kant’s science. Non-classical logics, like the fuzzy logic, challenges the traditional view.

  23. According to Popper, following Weyl, simpler theories or statements provide less parameters (see Popper 2005a, pp. 127–128). The falsifiability of an statement decreases with the number of parameters.

  24. Definitions and operations with fuzzy sets may be found in Zimmermann (1996, 2010).

  25. Popper defended a content measure of a theory and justified that Einstein’s theory has a greater content than Newton’s theory: “This makes Einstein’s theory potentially or virtually the better theory; for even before any testing we can say: if true, it has the greater explanatory power. Furthermore, it challenges us to undertake a greater variety of tests” (Popper 1994, p. 53).

  26. Ijjas et al. (2013, 2014) argue other problems in the inflationary mechanism like multiverse, unpredictability, and the trans-Planckian problem.

  27. Linde (2015) tries to answer this criticism.

  28. Metaphysical in Aristotelian sense, or at least in the sense of commentators on Aristotle, i.e., beyond nature (physis in Greek). Nature is empirical and as such it is thought of and intuited from space and time.

  29. See Ashtekar et al. (2006) for some proposals in a supposed quantum big bang.

  30. See Grier (2018) for an introduction to Kant’s antinomies.

  31. The second antinomy talks about the world and its parts, the third is about freedom, and the fourth antinomy of pure reason concerns God.

  32. Olbers’ paradox says an infinite and eternal universe would provide bright nights for us due to an infinite number of stars.


  • Abbott, B. P., LIGO Scientific Collaboration and Virgo Collaboration et al. (2016). Observation of gravitational waves from a binary black hole merger. Physical Review Letters, 116, 061102.

    Google Scholar 

  • Ade, P. A. R., Planck Collaboration et al. (2016). Planck 2015 results XIII. Cosmological parameters. Astronomy and Astrophysics, 594, A13.

    Google Scholar 

  • Al-Azm, S. J. (1968). Kant’s conception of the noumenon. Dialogue, 6(4), 516–520.

    Google Scholar 

  • Anderson, R. L. (1998). Truth and objectivity in perspectivism. Synthese, 115, 1–32.

    Google Scholar 

  • Ansoldi, S. (2007). Spherical black holes with regular center: A review of existing models including a recent realization with Gaussian sources. In Proceedings of BH2, dynamics and thermodynamics of black holes and naked singularities. Milano, Italy.

  • Ashtekar, A., Pawlowski, T., & Singh, P. (2006). Quantum nature of the big bang: An analytical and numerical investigation. Physical Review D, 73, 124038.

    Google Scholar 

  • Babich, B. (Ed.). (1999). Nietzsche, epistemology, and philosophy of science. London: Kluwer Academic Publishers.

    Google Scholar 

  • Bag, S., Sahni, V., Shtanov, Y., & Unnikrishnan, S. (2014). Emergent cosmology revisited. Journal of Cosmology and Astroparticle Physics, 07, 034.

    Google Scholar 

  • Barrau, A., Martineau, K., & Moulin, F. (2017). Seeing through the cosmological bounce: Footprints of the contracting phase and luminosity distance in bouncing models. Physical Review D, 96, 123520.

    Google Scholar 

  • Borde, A., Guth, A. H., & Vilenkin, A. (2003). Inflationary spacetimes are incomplete in past directions. Physical Review Letters, 90, 151301.

    Google Scholar 

  • Brandenberger, R., & Peter, P. (2017). Bouncing cosmologies: Progress and problems. Foundations of Physics, 47(6), 797–850.

    Google Scholar 

  • Carroll, S. M. (2001). The cosmological constant. Living Reviews in Relativity, 4, 1.

    Google Scholar 

  • Carroll, S. (2004). Spacetime and geometry: An introduction to general relativity. San Francisco: Addison Wesley.

    Google Scholar 

  • Coc, A., & Vangioni, E. (2017). Primordial nucleosynthesis. International Journal of Modern Physics E, 26(7), 1741002.

    Google Scholar 

  • Dorato, M. (2002). Kant, Gödel and relativity. In P. Gardenfors, K. Kijania-Placek & J. Wolenski (Eds.), Proceedings of the invited papers for the 11th international congress of the logic methodology and philosophy of science (pp. 329–346). Dordrecht: Synthese Library, Kluwer.

  • Einstein, A. (1916). Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 354(7), 769–822.

    Google Scholar 

  • Einstein, A. (1917). Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (pp. 142–152).

  • Ellis, G. F. R., & Maartens, R. (2004). The emergent universe: Inflationary cosmology with no singularity. Classical and Quantum Gravity, 21, 223–232.

    Google Scholar 

  • Freese, K. (2017). Status of dark matter in the universe. International Journal of Modern Physics D, 26, 1730012.

    Google Scholar 

  • Friedmann, A. (1922). Über die Krümmung des Raumes. Zeitschrift für Physik A, 10(1), 377.

    Google Scholar 

  • Gava, G. (2014). Kant’s definition of science in the Architectonic of pure reason and the essential ends of reason. Kant-Studien, 105(3), 372–393.

    Google Scholar 

  • Grier, M. (2018). Kant’s critique of metaphysics. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy (Summer 2018 Edition). Accessed 10 Aug 2019.

  • Guth, A. H. (1981). The inflationary universe: A possible solution to the horizon and flatness problems. Physical Review D, 23, 347–356.

    Google Scholar 

  • Hawking, S. W., & Ellis, G. F. R. (1973). The large scale structure of space–time. Cambridge: Cambridge University Press.

    Google Scholar 

  • Hawking, S. W., & Penrose, R. (1970). The singularities of gravitational collapse and cosmology. Proceedings of the Royal Society of London A, 314, 529–548.

    Google Scholar 

  • Ijjas, A., Steinhardt, P. J., & Loeb, A. (2013). Inflationary paradigm in trouble after Planck 2013. Physics Letters B, 723, 261.

    Google Scholar 

  • Ijjas, A., Steinhardt, P. J., & Loeb, A. (2014). Inflationary schism. Physics Letters B, 736, 142–146.

    Google Scholar 

  • Janiak, A. (2004). Kant as philosopher of science. Perspectives on Science, 12(3), 339–363.

    Google Scholar 

  • Kant, I. (1998). Critique of pure reason, translated by Paul Guyer and Allen W. Wood. Cambridge: Cambridge University Press.

  • Kant, I. (2004). Prolegomena to any future metaphysics, translated by Gary Hatfield. Cambridge: Cambridge University Press.

  • Leech, J. (2017). Kant’s material condition of real possibility. In M. Sinclair (Ed.), The actual and the possible: Modality and metaphysics in modern philosophy. Oxford: Oxford University Press.

    Google Scholar 

  • Lehners, J. L. (2008). Ekpyrotic and cyclic cosmology. Physics Reports, 465, 223.

    Google Scholar 

  • Lemaître, G. (1931). A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulæ. Monthly Notices of the Royal Astronomical Society, 91, 483.

    Google Scholar 

  • Linde, A. (2015). Inflationary cosmology after Planck 2013. In C. Deffayet, P. Peter, B. Wandelt, M. Zaldarriaga, & L. F. Cugliandolo (Eds.), Post-Planck cosmology: Lecture notes of the Les Houches Summer School: Volume 100, July 2013. Oxford: Oxford University Press.

    Google Scholar 

  • Merritt, D. (2017). Cosmology and convention. Studies in History and Philosophy of Modern Physics, 57, 41–52.

    Google Scholar 

  • Naraniecki, A. (2010). Neo-Positivist or Neo-Kantian? Karl Popper and the Vienna Circle. Philosophy, 85(4), 511–530.

    Google Scholar 

  • Neves, J. C. S. (2016). Are black holes in an ekpyrotic phase possible? Astrophysics and Space Science, 361, 281.

    Google Scholar 

  • Neves, J. C. S. (2017). Bouncing cosmology inspired by regular black holes. General Relativity and Gravitation, 49, 124.

    Google Scholar 

  • Neves, J. C. S. (2019a). Nietzsche for physicists. Philosophia Scientiæ, 23(1), 185–201.

    Google Scholar 

  • Neves, J. C. S. (2019b). Infinities as natural places. Foundations of Science, 24(1), 39–49.

    Google Scholar 

  • Neves, J. C. S., & Saa, A. (2014). Regular rotating black holes and the weak energy condition. Physics Letters B, 734, 44–48.

    Google Scholar 

  • Nietzsche, F. (2002). Beyond good and evil, translated by Judith Norman. Cambridge: Cambridge University Press.

  • Nietzsche, F. (2003). Writings from the late notebooks, translated by Kate Sturge. Cambridge: Cambridge University Press.

  • Nietzsche, F. (2005). The Anti-Christ, Ecce Homo and Twilight of the Idols, translated by Judith Norman. Cambridge: Cambridge University Press.

  • Novello, M., & Perez Bergliaffa, S. E. (2008). Bouncing cosmologies. Physics Reports, 463, 127–213.

    Google Scholar 

  • Palmquist, S. R. (1986). Six perspectives on the object in Kant’s theory of knowledge. Dialectica, 40(2), 121–151.

    Google Scholar 

  • Planck, M. (1900). Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum. Verhandlungen der Deutschen Physikalischen Gesellschaft, 2, 237.

    Google Scholar 

  • Plato. (1931). Timaeus, translated by B. Jowett. London: Oxford University Press.

  • Popper, K. (1994). Objective knowledge: An evolutionary approach. Oxford: Clarendon Press.

    Google Scholar 

  • Popper, K. (2005a). The logic of scientific discovery. London: Routledge Classics.

    Google Scholar 

  • Popper, K. (2005b). Unended quest: An intellectual autobiography. London: Routledge Classics.

    Google Scholar 

  • Pula, R. P. (1992). The Nietzsche–Korzybski–Sapir–Whorf hypothesis? ETC: A Review of General Semantics, 49(1), 50–57.

    Google Scholar 

  • Quintin, J., & Brandenberger, R. H. (2016). Black hole formation in a contracting universe. Journal of Cosmology and Astroparticle Physics, 11, 029.

    Google Scholar 

  • Robertson, H. P. (1935). Kinematics and world structure. Astrophysical Journal, 82, 284.

    Google Scholar 

  • Romero, G. E. (2013). Adversus singularitates: The ontology of space-time singularities. Foundations of Science, 18(2), 297–306.

    Google Scholar 

  • Starobinsky, A. A. (1980). A new type of isotropic cosmological models without singularity. Physics Letters B, 91, 99–102.

    Google Scholar 

  • Wald, R. M. (1984). General relativity. Chicago: The University of Chicago Press.

    Google Scholar 

  • Walker, A. G. (1937). On Milne’s theory of world-structure. Proceedings of the London Mathematical Society, 2–42(1), 90.

    Google Scholar 

  • Watkins, E., & Stan, M. (2014). Kant’s Philosophy of Science. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy (Fall 2014 Edition). Accessed 10 Aug 2019.

  • Weinberg, S. (2014). Cosmology. Oxford: Oxford University Press.

    Google Scholar 

  • Werkmeister, W. H. (1977). The critique of pure reason and physics. Kant-Studien, 68, 33–45.

    Google Scholar 

  • van den Berg, H. (2011). Kant’s conception of proper science. Synthese, 183, 7–26.

    Google Scholar 

  • Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.

    Google Scholar 

  • Zimmermann, H.-J. (1996). Fuzzy set theory and its applications. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Zimmermann, H.-J. (2010). Fuzzy set theory. WIREs Computational Statistics, 2, 317–332.

    Google Scholar 

Download references


This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Juliano C. S. Neves.

Ethics declarations

Conflict of interest

The author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Neves, J.C.S. Proposal for a Degree of Scientificity in Cosmology. Found Sci 25, 857–878 (2020).

Download citation

  • Published:

  • Issue Date:

  • DOI:


  • Big bang
  • Singularity
  • Kant
  • Nietzsche
  • Popper
  • Fuzzy sets
  • Degree of scientificity
  • Standard model
  • Emergent cosmologies
  • Bouncing cosmologies