Abstract
The sum of all objects of a science, the objects’ features and their mutual relations compose the reality described by that sense. The reality described by mathematics consists of objects such as sets, functions, algebraic structures, etc. Generally speaking, the use of terms reality and existence, in relation to describing various objects’ characteristics, usually implies an employment of physical and perceptible attributes. This is not the case in mathematics. Its reality and the existence of its objects, leaving aside its application, are completely virtual and yet clearly organized. This organization can be recognized in the creation of axioms or in the arrival at new theorems and definitions. It results either from a formalization of an intuitive idea or from a combinatorics that has not been guided by intuition. In all four possible cases—therefore, also in the two in which there is no intuitive “lead”, we can plausibly talk about a discovery of mathematical facts and thus support the Platonist view.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Tieszen (2011, p. 33).
When we speak about the basic formal conditions fulfilled by a set of axioms in a theory, we mean consistency, completeness, independence. Normally, these are standard propositions/requirements of every set of axioms. In this text, we will refer to a set of axioms (and the theory derived from it) as being consistent if it does not allow for two contradictory propositions to be derived from it. Consistency thus defined can also be found in reference books under the sintagm syntatical consistency, as opposed to semantical consistency, which is fulfilled if a model for the theory can be provided (Balaguer 1998, p. 70). Independence of a set of axioms implies impossibility to derive any new axiom from the remaining ones. Finally, we say that a set of axioms is complete if every proposition within the theory built upon that set can be proved or disproved.
Hyperbolic geometry.
Borceux (2014, pp. 90, 106, 213, 338).
Transfinite numbers.
Cantor (1883).
This Frege’s principle will later be firther developed in Wittgenstein’s Philosophical Investigations.
Tait (2005, pp. 90).
For instance, the historical beginnings of geometry cannot be established with certainty, whereas the origins of some other disciplines are more recent. Complex analyses, for example, was born in the nineteenth century.
Topology, cryptography, numerical analysis, game theory, etc.
For instance, we still do not have a proof for consistency of the Zermelo–Fraenkel axioms system (ZFC).
In any formal system adequate for number theory there exists an undecidable formula, that is, a formula that is not provable and whose negation is not provable.
If the formal theory is consistent, then its consistency cannot be proved within the formal theory. For further information on Gödel’s theorems see Wang (1995).
By this we refer to the function whose domain and codomain is set R.
Gödel (1983, pp. 456–457).
Maddy (1990, p. 59).
Maddy (2005).
Tait (2005).
In the case of geometry.
Pinter (1982, pp. 109–111).
Meno, 82, 85b.
Hersh (1997, p. 138).
Iversen (1992, p. 57).
Szabo (1969, pp. 33–37).
For example: A point is that of which there is no part; A line is a length without breadth; etc.
For example: One can draw a straight line from every point to every other point; one can produce a segment continuously in a straight line; etc.
Sometimes we can find several axiomatizations for one theory. If for one axiomatization of a theory T, A1 represents a set of basic propositions (axioms) and B1 a set of deduced propositions, and for another axiomatization of the same theory T, A2 represents a set of basic propositions (axioms) and B2 is a set of deduced propositions, then A1 ≠ A2, and B1 ≠ B2, because this is the case of different axiomatizations. However, since these are axiomatizations of one and same theory, then A1 ∪ B1 = A2 ∪ B2. In other words, a set of all truths made of sets of all basic and derived propositions is the same in both axiomatizations, but the basic proposition in the first axiomatization can be the derived proposition in the second, and the basic proposition in the second can be the derived in the first axiomatization. For further information, see, for example, Tarski (1994, pp. 112, 113, 131).
Maddy (1990).
When, in the further text, we discuss various axiomatizations, we will regard them as sets of axioms determining various theories. In other words, if A1 and A2 represent different sets of axioms, then we will regard them as not determining the same theories.
It is doubtless that there are constructions of certain spaces which will never be formally identified and axiomatically established. They will exist always as a possibility.
If a straight-line falling across two (other) straight-lines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straight-lines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side).
A large number of authors throughout history have investigated whether the Euclid’s Fifth Postulate ir really an elementary proposition, or whether it can be derived from other postulates. Its complexity, when compared to the rest of postulates, has attracted special attention. It was not simple and intuitively acceptable as other postulates, which opened up a space for research and debates regarding its possible substitution by another elementary proposition. For more information, see Trudeau (1987).
At this point it would be useful to make a historical parallel in order to further the argument. Not a single respectable historian can dispute Columbus’s discovery of America, even though his arrival at the new continent was a mere accident. The discovery of the new continent was a result of ideas, intentions, desires and combinatorics from which an entirely different outcome was expected—arrival at India.
See, for instance, Aczel (1996).
See Faltings (1995).
Using the associative law for summing up natural numbers, we can put the above sum as follows:
$$\begin{aligned} & \left( {1 + \left( {2n{-} \, 1} \right)} \right) + \left( {3 + \left( {2n{-}3} \right)} \right) + \cdots + \left( {\left( {n{-}1} \right) + \left( {n + 1} \right)} \right) {\kern 1pt} = 2n + 2n + \cdots + 2n = 2n*\left( {n/2} \right) = n^{2} , \\ \end{aligned}$$in the case when n is an even number. The similar process can be used when n is an odd number.
References
Aczel A (1996) Fermat’s last theorem. Delta, New York
Balaguer M (1998) Platonism and anti-platonism in mathematics. Oxford University Press, Oxford
Borceux F (2014) An axiomatic approach to geometry. Springer, London
Cantor G (1883) Über unendliche, lineare Punktmannigfaltigkeiten, 5. Math Ann 21(5):545–586
Faltings G (1995) The proof of Fermat’s last theorem by R. Taylor and A. Wiles. Not Am Math Soc 42(7):743–746
Gödel K (1983) Russell’s mathematical logic. In: Benacerraf P, Putnam H (eds) Philosophy of mathematics. Cambridge University Press, Cambridge, pp 447–469
Hersh R (1997) What is mathematics, really?. Oxford University Press, Oxford
Iversen B (1992) Hyperbolic geometry. Cambridge University Press, Cambridge
Maddy P (1990) Realism in mathematics. Oxford University Press, New York
Maddy P (2005) Mathematical existence. Bull Symb Logic 11(3):351–376
Pinter C (1982) A book abstract algebra. McGraw-Hill, New York
Szabo ME (ed) (1969) The collected papers of Gerhard Gentzen. North-Holland, Amsterdam
Tait W (2005) The provenance of pure reason. Oxford University Press, Oxford
Tarski A (1994) Introduction to logic and to the methodology of the deductive sciences. Oxford University Press, Oxford
Tieszen R (2011) After Gödel. Oxford University Press, Oxford
Trudeau RJ (1987) The non-Euclidean revolution. Birkhäuser, Boston
Wang H (1995) Reflections on Kurt Gödel. The MIT Press, Cambridge
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Drekalović, V. Some Aspects of Understanding Mathematical Reality: Existence, Platonism, Discovery. Axiomathes 25, 313–333 (2015). https://doi.org/10.1007/s10516-014-9253-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10516-014-9253-8