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. 159-160My 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. 160Many 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. 161One 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. 162Frege 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. 165Lakatos’ 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. 166Lakatos (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. 166The 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. 173Are 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. 174Like 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