Objects of mathematical discourse - Chapter 6 - Objects of mathematical discourse

Aug 22, 2026

Chapter 6 explores what “mathematical objects” are through the lens of mathematical discourse and mathematizing. It frames objects as discursive constructs rather than entities existing independently, emphasizing that mathematical activity generates ideal objects such as numbers and geometric figures. The chapter introduces “realizations” and a “realization tree” that links symbols, notations, and perceptually grounded instances within a discourse. It argues these structures are personal, yet shaped by public discourses, and that transitions among realizations underpin mathematical problem solving.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages30

Reading Time1 h 16 min

AccessFree to download

CHAPTER 6 Objects of mathematical discourse: What mathematizing is all about

1. Mathematical objects

1.1 Discursive objects

²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·²·² CHAPTER 6 Objects of mathematical discourse: What mathematizing is all about I close my eyes and see a flock of birds. The vision lasts a second, or perhaps less; I am not sure how many birds I saw. Was the number of birds definite or indefinite? The problem involves the existence of God. If God exists, the number is definite, because God knows how many birds I saw. If God does not exist, then the number is indefinite, because no one can have counted. In this case I saw fewer than ten birds (let us say) and more than one, but did not see nine, eight, seven, six, five, four, three or two birds. I saw a number between ten and one, which was not nine, eight, seven, six, five, etc. That integer--not-nine, not-eight, not-seven, not-six, not-five, etc,--is inconceivable. Ergo, God exists. Luis Jorge Borges1 I remember as a child, in fifth grade, coming to the amazing (for me) realization that the answer to 134 divided by 29 is 134/29 (and so forth) . What a tremendous labor-saving device! To me, ‘134 divided by 29’ meant a tedious chore, while 134/29 was an object with no implicit work. I went excitedly to my father to explain my discovery. He told me that of course this is so, a/b and a divided by b are just synonyms. To him, it was just a small variation in notation. William Thurston2 The ‘content’ of mathematics does not exist in the material world; it is created by the activity of mathematics itself and consists of ideal objects like numbers, square roots and triangles. Michael A. K. Halliday3 Mathematicians and philosophers have been grappling with the idea of a mathematical object for ages, always recognizing its inherent blurriness, but never considering the option of simply giving it up. After all, if there is no such thing as mathematical reality, why should one bother to engage in mathematical investigations? In their most extreme forms, the claims about the nature of mathematics implied that mathematical objects have an independent existence of sorts. Those who objected, have been reproached by their Platonically minded colleagues: Everything considered, mathematicians should have courage of their most profound convictions and thus affirm that mathematical forms indeed have an existence that is independent of the mind considering them 4 If I opt for operationalizing the time-honored idea of mathematical object rather than trying to do without it, it is only partly out of reverence to its long history, and certainly not because of any Platonic leanings on my part. My main reason is the hope that this special notion, with its deep metaphorical roots, will help us in understanding the developmental connection between mathematical discourses and discourses on material reality. Ch 6 – objects Last printed 6/3/21 4:13:00 PM 1. Mathematical objects 1.1 Discursive objects While mathematizing, we are in the incessant chase after the objects of our activity. True, in this “object hunt” we proceed from one tangible entity to another, but I called these latter entities "realizations" rather than "mathematical objects." There is a number of reasons for this lexical restrain. First, realizations are characterized by being perceptually accessible – a property which one does not expect to find in a genuine mathematical object. Second, one signifier would usually have many visual realizations and it would be difficult to tell which of them deserves being singled out as "the" object. Finally, as was already mentioned, the distinction between signifier and realization is relative. Symbolic artifacts are often exchangeable in these two roles. For example, one can use a table of function values as a signifier and realize it in a formula, and vice versa – the formula may be realized in a table. Thus, whether a word, algebraic symbol or icon should count as asignifier or as a realization of a signifier is a matter of use, not of any intrinsic property of

FAQs

Why does the chapter distinguish between “realizations” and “mathematical objects”?

“Realizations” are perceptually accessible and multiple visual instances typically correspond to one signifier. The chapter also treats the signifier–realization distinction as relative and based on use rather than intrinsic properties.

What is a “realization tree,” and how is it used to define discursive objects?

A realization tree organizes how a signifier leads to nested realizations. In a given discourse, the discursive object signified by S is defined as the realization tree of S within that discourse.

Are mathematical objects personal or public according to this chapter?

The chapter states that realization trees—and thus mathematical objects—are personal constructs, even though they originate in public discourses that support only certain versions of such trees.

Aug 22, 2026

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