# What characteristics define something as a mathematical entity?

It is easy to give examples of mathematical entities: Natural numbers, geometrical figures, sets, functions of variables ranging over numerical sets, etc. The list seems endless. Yet, listing those entities which are objects of known mathematical theorems would presumably not do justice to the mathematical idea of mathematical entity.

So, what characteristics if any truly qualify something as a mathematical entity?

Thanks for scholarly references.

• Charles Parsons, Mathematical Thought and its Objects (Cambridge UP, 2008) Commented Mar 5 at 9:39
• That is a lovely question. One that is easy to understand, demands no specialist knowledge, and requires nothing but time to think. Commented Mar 5 at 9:44
• None. Mathematical entities are placeholders in mathematical structures, that is systems described precisely enough to allow proving claims about them deductively. Their characteristics as entities are orthogonal to what makes them mathematical, namely, the formality of description. Any abstract entity can be made mathematical by formalizing it (in modern approximation, defining it in ZFC). Already Aristotle points towards such operational view of mathematics, see Cadavid, Existence in a way: resolving a tension in Aristotle’s philosophy of mathematics. Commented Mar 5 at 11:54
• See Quine “Any progression will serve as a version of number so long as and only so long as we stick to one and the same progression. Arithmetic is, in this sense, all there is to number: there is no saying absolutely what the numbers are; there is only rithmetic” (Ontological Relativity, 1968). Commented Mar 5 at 13:49
• Where did you get the idea listing those entities which are objects of known mathematical theorems would not do justice to the mathematical idea of mathematical entity? Commented Mar 6 at 1:02

If you're looking for a real definition of mathematical entity that goes beyond a general purpose dictionary, then you are engaged in one of the central question of the philosophy of mathematics (SEP). Generally, mathematical entities are taken to possess the characteristics of abstract objects (SEP). From SEP:

One doesn’t go far in the study of what there is without encountering the view that every entity falls into one of two categories: concrete or abstract. The distinction is supposed to be of fundamental significance for metaphysics (especially for ontology), epistemology, and the philosophy of the formal sciences (especially for the philosophy of mathematics);

In particular, mathematical abstract objects are marked by a number of characteristics that form the basis for claims about their philosophical basis. An accessible book talking about what constitutes mathematical entities is Linnebo's Philosophy of Mathematics (GB). A look at the table of contents includes some clues of various perspectives. According to famous thinkers, mathematical entities:

1. Have logical foundations
2. Have numerosity
3. Arise from intuition
4. Possess structure
5. Are conveyed by language
6. Are formal but have meaning
7. Can be related to empirical experience

A simple example, like the triangle, can help to make this clear:

1. We can reason about triangles according to Euclid's axioms, for instance.
2. A triangle has three, straight sides.
3. We intuitively grasp that some shapes are triangle-like, for instance, even a shape where the three sides are slightly curved. We can intuit a triangle has three sides and count to three.
4. Triangles can be represented by sets of ordered pairs, or matrices if we prefer.
5. We can talk at length about triangles, and recognize that triangles exist in some sense independent of the physical world.
6. We can use formalisms like taxonomies of triangles and discuss the theory of triangles, trigonometry as a metatheory.
7. Triangles are useful in that they are used in building trusses and pyramids and all manner of things in the physical world.
• "Have logical foundations" Presumably, this means that they cannot have mutually contradictory characteristics, yes? - 2. Don't all the characteristics listed in your answer apply to abstract entities in general, including ones which are not considered by mathematicians? Commented Mar 6 at 10:56
• @Speakpigeon The relationship between logic and mathematical systems can be complicated. Brouwer, the father of rejected two classical laws of logic in his mathematics when he developed intuitionistic logic (SEP). But to simplify, generally numbers and shapes don't have contradictory properties. Circles aren't squares, primes aren't composites. The second question is quite interesting.
– J D
Commented Mar 6 at 14:13
• I would suggest that mathematical structure is understood as being capable of describing all abstract objects, in some way or another. A schematic for an electric circuit composed of transistors can have a mathematical description in Boolean logic. A architectural plan is composed of geometric shapes and arithmetic measurements. Zipf's Law can be applied to natural language corpora. If you read the SEP article Abstract Object, you'll see that there's no standard way of visiting it. My preferred way of demarcating mathematical objects as distinct from a more general class:
– J D
Commented Mar 6 at 14:21
• Mathematical objects are fictions that are intuitively constructed regarding experiences of quantities, collections, shapes, relations, operations, directions, and truth values. But YMMV with my schema.
– J D
Commented Mar 6 at 14:25

The fundamental characteristic of a mathematical entity is the requirement of non-contradiction. If there is a contradiction built in, the entity will never fly as a mathematical one. Such a requirement applies far broader than "an entity in an axiomatic system" because the latter is a fairly modern idea, whereas the principle of non-contradiction was already viewed as a sine qua non of a mathematical entity by Leibniz. Here non-contradiction did not necessarily mean to Leibniz that some objectification can be found for the entity in physical reality. On the contrary, Leibniz emphasized a distinction between absolute impossibility (equivalent to contradiction) and accidental impossibility (something not found in the physical realm but nonetheless useful as a tool in mathematics). Examples of the latter category to Leibniz are negatives, imaginary roots, and infinitesimals (as well as their inverses: infinite quantities). For a detailed study of Leibniz's take on mathematical entities, see our recent article

Katz, M.; Kuhlemann, K.; Sherry, D.; Ugaglia, M. "Leibniz on bodies and infinities: rerum natura and mathematical fictions." Review of Symbolic Logic 17 (2024), no. 1. https://doi.org/10.1017/S1755020321000575, https://arxiv.org/abs/2112.08155

• "The fundamental characteristic of a mathematical entity is the requirement of non-contradiction": wrong, logic is the ruleset that determines consistency (ergo, contradictions). Mathematical objects are sustained on logic, but they are not the same. Commented Mar 5 at 15:42
• @RodolfoAP, I think you may have missed my point. The Leibnizian principle of non-contradiction was developed long before logic existed as a separate discipline (though Leibniz's genius may have contributed to its eventual creation). Arguably, the principle of non-contradiction is more basic than any talk about logic as a foundation for mathematics (a view that may not even have been Leibniz's, though it is a common view today). Commented Mar 5 at 15:50
• @MikhailKatz "the requirement of non-contradiction" If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist. So, for example, a real number whose square is -l does not exist mathematically. But if it can be proved that the attributes assigned to the concept can never lead to a contradiction by the application of a finite number of logical processes, I say that the mathematical existence of the concept (for example, of a number or a function which satisfies certain conditions) is thereby proved -- David Hilbert, 1900 Lecture in Paris Commented Mar 5 at 16:31
• @MichaelCarey, I don't think Leibniz meant to claim that one can delineate precisely which the noncontradictory notions are. He is merely claiming that if a notion has been shown to be contradictory, then one can no longer use it anymore. Both Leibniz and Hilbert were aware of the fact that catching a contradiction lurking in the background could be very tricky and is not necessarily obvious. I don't think Goedel's incompleteness results are that relevant here, even though it obviously was and remains a fabulous accomplishment. To take a realistic example: if a certain large cardinal ... Commented Mar 6 at 15:39
• ... hypothesis is proven to be contradictory, this does not mean that all of modern mathematics therefore collapses. It just means that we don't use that particular hypothesis, in an ardent hope that the rest of set theory will not follow suit :-) Commented Mar 6 at 15:40

First of all, a mathematical entity is an idea, not a physical thing.

Secondly, a mathematical idea is an undefinded fundamental concept within an axiomatized theory or it has a precise definition within the theory.

As an example, “set” is an undefined fundamental concept of set theory, while the “intersection of two sets” is defined within set theory. The relation between the undefined fundamental concepts and the definitions within a formalized theory is fixed by the axioms of the theory. Introducing new mathematical concepts into a theory should be consistent with the concepts already at hand.

The examples of the entity “set” or more specific concepts like “topological space”, “topos”, “scheme” show that there is nearly no restriction of the content of a mathematical entity.

• "a mathematical entity is an idea" Quine objected. - 2. "a mathematical idea is an undefinded . . . concept within an axiomatized theory" So not really "within" the theory. - 3. "“set” is an undefined . . . concept of set theory, while the “intersection of two sets” is defined within set theory" How can the notion of the intersection be properly defined in the theory if the notion of set itself isn't? We don't care what we are talking about? We don't know? Mathematical theories perhaps are not self-sufficient? - 4. "there is nearly no restriction" What, anything could be? Commented Mar 5 at 11:12
• @Speakpigeon 1. How does Quine object, can you name a precise reference? 2. Of course is the concept "set" a concept within(!) set theory. 3. "intersection" is defined relatively to the undefined fundamental concepts "set" and "element". Commented Mar 5 at 11:28
• In On what there is, Quine discusses the case of Pegasus and of the Parthenon and explains that we shouldn't confuse the thing and our idea of it. - 2. "the concept "set" a concept within(!) set theory" Ok, so it is a meaningless concept within the theory. - 3. ""intersection" is defined relatively to the undefined . . . concepts "set" and "element"" Ok, so it is also a meaningless concept within the theory. Thanks. Commented Mar 5 at 11:36
• IMO Quine confirms that some ideas have a referent and others do not: Pegasus is an idea without referent. The issue is not "idea versus no idea" but "referent versus no referent". Many mathematical concepts do not have a referent, see my examples. Commented Mar 5 at 11:47
• "Quine confirms that some ideas have a referent" Of course and this is not the debate. My point was that Quine would have objected to your claim that a mathematical entity is an idea. If a geometrical figure for example is a mathematical entity, then a triangle is a mathematical entity, while the idea of the triangle is not. - 2. "* The issue is not "idea versus no idea"" Don't be silly. This is not what I said. - 3. "*Many mathematical concepts do not have a referent, see my examples." Where? Commented Mar 5 at 16:25

Here are a few quotes from Modern Science, Metaphysics, and Mathematics by Heidegger regarding mathematics. (36 pages, Internet Archive)

We attain this fundamental feature of modern science for which we are searching by saying that modern science is mathematical. (page 249)

Therefore, we must now show in what sense the foundation of modern thought and knowledge is essentially mathematical. (page 254)

The mathematical project of Newtonian bodies leads to the development of a certain "mathematics" in the narrow sense. The new form of modern science did not arise because mathematics became an essential determinant. Rather, that mathematics, and a particular kind of mathematics, could come into play and had to come into play is a consequence of the mathematical project. The founding of analytical geometry by Descartes, the founding of the infinitesimal calculus by Newton, the simultaneous founding of the differential calculus by Leibniz—all these novelties, this mathematical in a narrower sense, first became possible and above all necessary on the grounds of the basically mathematical character of the thinking. (page 269)

In the essence of the mathematical, as the project we delineated, lies a specific will to a new formation and self-grounding of the form of knowledge as such. (page 272)

• Where does Heidegger now unveil what "the mathematical character of thinking" is? Commented Mar 5 at 12:22
• @JoWehler Maybe in the discussion of Descartes on page 275 : "all this happened in the midst of a period in which, for a century, mathematics had already been emerging more and more as the foundation of thought and was pressing toward clarity." . . . "this means that the mathematical wills to ground itself in the sense of its own inner requirements. It expressly intends to explicate itself as the standard of all thought and to establish the rules which thereby arise." Commented Mar 5 at 12:30
• I cannot become friends with Heidegger. I see him only beating around the bush. How much did he understand how mathematics works? Commented Mar 5 at 14:35
• @JoWehler He seems to start mainly from the axioms of Descartes' Regulae ad directionem ingenii (Rules for the Direction of the Mind) and the Mathesis universalis. This wider sense of mathematics makes sense in the context of Penrose's math-matter-mind triangle, with maths as formal knowledge. Commented Mar 5 at 14:52
• @ChrisDegnen How is that an answer to my question? - 2. "modern science is mathematical" So no science is "modern" if it is not mathematised? Isn't mathematisation rather just a possible stage in the development of a science? Perhaps even one which is not always possible? Commented Mar 5 at 17:18