Abstract
On the face of it, physics and mathematics are about different things. The objects of physics are those which we causally interact with every day and which give us our sensations of heat, light, texture, smell and sound. Physicists assume their existence and try to discover their properties and how they cause the experiences that they do. In contrast, the objects of mathematics are usually deemed to be irrelevant to the causal nexus. If mathematical objects exist at all, then they exist outside of space and time in an eternal, unchanging state of pure being. We can’t interact with them in a sensual way, only by pure intellection, and any such interaction defies the laws of physical objects because it does not involve any interchange of energy, or any change at all in the grasped object, merely a change in our own mental state.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Note that Tarski and Corcoran show that our logic is a logic of cardinal number:
It turns out that the only properties of classes (of individuals) which are logical are properties concerning the number of elements in these classes. That a class consists of three elements, or four elements … that it is finite, or infinite-these are logical notions, and are essentially the only logical notions on this level. (Tarski and Corcoran 1986: 151)
- 2.
I shall speculate on what happens at limit ordinals in the section on laws (Sect. 7.2.3.3 below). Roughly speaking, the suggestion is that monads at a given level of the hierarchy develop an ever-improving model of the concepts and objects in their world and, at the limit ordinal, that model is encoded as an ideal Pythagorean theory of laws which starts the next level of the hierarchy.
- 3.
As an aside, considering topics for future research, Peirce’s presentation of thought as a dialogue between minds links his semiotics to the fascinating area of semantic games in logic (e.g. Hintikka’s idea that any rational human activity can be played via logical games, and the computational idea of meaning as a dialogue between the system and the environment). Furthermore, if we restrict our attention to the computational (i.e. non-semantic) aspect of thought processes, then two-player games are used extensively in mathematics (e.g. to represent results in set theory such as the Borel determinacy theorem). This may provide some insight into the connection between bare mathematics and full mathematics.
References
Burch, R. 2014. Charles Sanders Peirce. In The Stanford Encyclopedia of Philosophy (Winter ed.), edited by Edward N. Zalta, downloaded from http://plato.stanford.edu/archives/win2014/entries/peirce/
Hohwy, J. 2013. The Predictive Mind. Oxford: Oxford University Press.
Johnston, M. 2009. Saving God: Religion after Idolatry. Princeton: Princeton University Press.
Kant, I. 1781. Critique of Pure Reason. Translated by Paul Guyer and Allen Wood. Cambridge: Cambridge University Press, 1998.
McDowell, J. 1994. Mind and World. Cambridge, MA: Harvard University Press.
Nelson, E. 2011. Warning Signs of a Possible Collapse of Contemporary Mathematics. In Infinity: New Research Frontiers, edited by M. Heller and W.H. Woodin, 76–88. New York: Cambridge University Press.
Tarski, A., and J. Corcoran. 1986. What Are Logical Notions? History and Philosophy of Logic 7(2): 143–154.
Tieszen, R. 2000. Gödel and Quine on Meaning and Mathematics. In Between Logic and Intuition: Essays in Honor of Charles Parsons, edited by G. Sher and R. Tieszen, 232–256. Cambridge: Cambridge University Press.
Wang, H. 1996. A Logical Journey: From Gödel to Philosophy. Cambridge, MA: MIT Press.
Wheeler, J. 1990. A Journey into Gravity and Spacetime. New York: W.H. Freeman.
Wheeler, J. 1996. At Home in the Universe. New York: Springer-Verlag.
Wigner, E. 1960. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications of Pure and Applied Mathematics 13: 1–14.
Wilczek, F. 2006. Reasonably Effective: I Deconstructing a Miracle. Physics Today 59: 8–9.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 The Author(s)
About this chapter
Cite this chapter
McDonnell, J. (2017). Conclusion. In: The Pythagorean World. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-40976-4_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-40976-4_7
Published:
Publisher Name: Palgrave Macmillan, Cham
Print ISBN: 978-3-319-40975-7
Online ISBN: 978-3-319-40976-4
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)