Representational Innovation and Mathematical Ontology by Madeline Muntersbjorn

I suggest there are such
things as mathematical objects and, further, we really do know, with varying
degrees of precision, what they are. Does a robust commitment to
mathematical realism commit us to the existence of immutable and eternal objects? Not necessarily. Indeed, I suspect that mathematical objects are
real but emergent phenomena. As such, they may share many properties
in common with other natural phenomena that come into existence
gradually, like organic kinds. For example, some mathematical objects
(counting numbers) may be older than others (imaginary numbers). An
emergentist view of mathematical objects frees us from having to choose
between epistemological inadequacy and a loss of mathematical content.
The metaphysics I envision would explain not only the nature of mathematical
objects, but also the ontological instability characteristic of the
growth of mathematics. p. 159-160

My critique of presentism ends with an endorsement of naturalism.
Inspired by Frege and Lakatos, I propose a naturalistic approach to the
study of mathematics over time that connects representational innovations
directly with the emergent reality of mathematical objects. p. 160

Many insist falsely that we must choose between two alternatives:
Mathematics must be either created or discovered. However, I
reject both alternatives. Instead, I propose that mathematics is cultivated. p. 161

One may be a realist about mathematical objects without subscribing
to the view that the existence of these objects is wholly independent of the
symbolic means via which they emerge. … I suggest that the existence of mathematical objects
requires the codification of explicit notational devices to make them
not only manifest but also real. … Systems
of notation entail ontological inferences. p. 162

Frege did not invent the concept-script to
name extant mathematical objects and make them easier to manipulate.
Rather, he was impelled by “a necessity inherent in the subject matter
itself” (p. 7). My historiography of mathematics seeks to understand the
nature of this necessity. … Conclusions of mathematical inferences are “contained” in their
premises “as plants are contained in their seeds, not as beams are contained
in a house” (ibid., p. 101).p. 165

Lakatos’ philosophy of mathematics cannot
be identified with historicism, fictionalism, or any socio-constructivist
account of mathematics that places mathematicians wholly in charge,
conforming their discipline in accordance their inclinations. p. 166

Lakatos (1976) invokes organic imagery:
Mathematical activity is human activity.. . . Mathematics, this product of human activity,
‘alienates itself’ from the human activity which has been producing it. It becomes a living,
growing organism that acquires a certain autonomy from the activity which has produced
it; it develops its own autonomous laws of growth, its own dialectic. The genuine creative
mathematician is just a personification, an incarnation of these laws which can only realise
themselves in human action. (p. 146) p. 166

The insight Frege and Lakatos both seek to call to our attention is
simply that, “in an important sense, mathematics generates its own content”
(Kitcher 1988, pp. 313–314). p. 166

On that one, Barany and Mackenzie might disagree.

One particularly promising “law” of mathematical
development has been identified as an important heuristic, namely, make
the implicit explicit.10 See Lakatos (1976, p. 50)

I offer the following sketch as a
stimulant for further inquiry. Mathematical objects begin as tacit insights
or, more fancifully, hypothetical residents of the imagination visible only
to the mind’s eyes of uniquely talented individuals. They acquire concrete
representation as the need arises in particular problem spaces. … Initially, representational innovations are
intended to make implicit procedures explicit. They are not intended to rename
mathematical objects, but are intended to represent modus operandi
on extant mathematical objects. Sometimes problem-solving strategies
employing these novel notations are so successful that their success becomes
an explanandum. Concrete symbols refer to emergent objects when
the existence of these objects is the only satisfactory explanation of the
problem-solving success of the strategy. These explanations are articulated
explicitly by mathematicians who seek to increase the scope of their inquiry.
Mathematical objects do not exist before the introduction of new
mathematical symbols, but only come into existence after systems of signification
capable of referring to them are codified via the mathematical
community. p. 173

Are mathematical objects created or discovered? This question is
based on the false metaphysical premise that there are two kinds of reality,
the created reality, dependent on the mind, and the discovered reality,
independent of the mind. Many things would not exist were it not for the
deliberate habits of generations of mindful beings. Domestic felines and
brewer’s yeast are two of my favorites. These organic kinds are characterized
by an autonomous necessity over which we exert little control, as
any home brewer or cat fancier will confirm. Were these kinds created or
discovered? Of course, neither answer is correct. Cats and beer are the
result of gradual processes of causal interactions between nature and nurture. As such, they transcend the creation-discovery distinction. Similarly,
mathematical objects are neither created nor discovered. Instead, I propose
that they emerge from the cultivation of mathematics. p. 174

Like agrarians, we cultivate mathematics.
Sometimes, when the harvest is brought in, we are surprised by what our
efforts yield. However, this surprise is not a consequence of our ignorance
of what was “there all along” to be uncovered by the patient elimination
of misleading distractions. Rather, we are surprised when the fruits of our
labor are both startling and new. p. 175




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