“The argument is carried out not in mathematical symbols but in ordinary English, there is no obscure or technical terms. Knowledge of calculus is not presupposed. In fact, one hardly need to know how to count. Yet any mathematician will immediately recognize the argument as mathematical.”
Test your intuition 44: what is the argument? who and where wrote this view about “what is mathematics”.
Zero-knowledge answers please.
Comments regarding this view itself and on “what is mathematics” are also welcome.
(Here are other posts on “What is mathematics.”)
PS. The last facetious sentence was omitted in the Journal version of the paper. (Indeed it was a good decision to take it out.) PPS Yannai Gonczarowski pointed out the the journal formulation is also rather condescending (perhaps even more so) towards non-mathematicians.
Combinatorics and more 
The proof of the existence of infinitely prime numbers.
Mathematics is best compared to a combination of masochism and opium:
an addiction to the endorphines released after the pain fades from certain self-imposed mental tortures.
And like other addicts, Mathematicians tend to rationalize their indulgences: as “work” on some “important problem”.
Thus the inherent impossibility to “define” Mathematics: YKINMK.
A high school teacher of mine (with a PhD in mathematics) once jokingly gave me this true, but also not very satisfying definition: “Mathematics is what mathematicians do.”
If mathematicians are the only people, who can perceive arguments that require more than two steps, then the world is in a very dire state of affairs. Maybe it is. Most likely it is right in saying that mathematical arguments are arguments with sufficient precision. It sounds about right. Or at least it explains why mathematicians spend quite some time arguing about what is sufficient precision.
My view (with respect to applied mathematicians) is that mathematics is the branch of natural philosophy that concerns itself with only making true statements.
This involves tradeoffs. Most visibly, we gave up our right to say the RH is true (to pick an example) even though it’s been verified as rigorously as almost any law of physics or result in biology. Less visibly, our statements are always conditional (“if A then B”) — we always need axioms. But we also get benefits: our literature largely doesn’t grow stale: results in our literature have half-lives orders of magnitude greater than in the sciences.
I know that this definition sidelines the Grothendieck school emphasis on definitions as opposed to theorems, but I still think it best captures the spirit of pure mathematics.
Lease A Glyph
I would say it’s Coase’s Theorem, except that that is really a result in economics, not mathematics.
Ronald Coase (1960) only gave simple numerical examples based on fascinating historical examples. At Chicago, his departmental colleagues found it too counterintuitive to believe, more of a “what is wrong with this reasoning” puzzle Coase had foisted on them, until finally Milton Friedman had an aha! moment and then convinced everyone else.
I know this one! 🙂 Heard it in a math talk even. It’s about the Gale-Shapley algorithm for the stable marriage problem, if I recall correctly.
This was the talk: https://personal.utdallas.edu/~nxw170830/docs/Presentations/kjw_hawaii_match.MP4. See around the 16 minute mark for where the quote appears.
A strong wind and an insurance company
Gale and Shapley
Stable Marriages?
It was mentioned in the AMS notices or the monthly in the past few years.
Without attempting to address the issue in its broader dimensions, I would suggest that mathematics ought to be considered as a set of precise, symbolic, languages that serves as a lingua franca for the physical sciences by representing and communicating physical phenomena unambiguously on the basis of evidence-based reasoning (in the sense of reference 1).
1. In other words:
(a) Any language of such a set, say the first order Peano Arithmetic PA (or Russell and Whitehead’s PM in Principia Mathematica, or the Set Theory ZF) is, ideally, intended to adequately express and/or effectively communicate—in a finite and unambiguous manner—relations between elements that are external to the language PA (or to PM, or to ZF).
(b) Moreover, each such language is two-valued if we assume that there is, again ideally, some evidence-based methodology (effective method) to finitarily decide whether a specific relation can be said to hold (be true) or to not hold (be false) externally under any well-defined interpretation of the language.
2. Further:
(a) A selected, finite, number of primitive formal assertions about a finite set of selected primitive relations of a language, say L, are defined as axiomatically L-provable;
(b) All assertions about relations that can be effectively defined in terms of the primitive relations are termed as L-provable if, and only if, there is a finite sequence of assertions of L, each of which is either a primitive assertion or which can effectively be determined in a finite number of steps as an immediate consequence of any two assertions preceding it in the sequence by a finite set of finitary rules of consequence;
(c) All L-provable relations interpret as true under any well-defined interpretation of L.
3. To place this view of mathematics in perspective, I find it helpful to make an—admittedly arbitrary—distinction between:
(a) The Natural Scientist’s hat: whose wearer’s responsibility is recording—as precisely and as objectively as possible—our sensory observations (corresponding to computer scientist David Gamez’s `Measurement’) and their associated perceptions of a `common’ external world (corresponding to Gamez’s `C-report’, and to what some cognitive scientists, such as Lakoff and Nunez, term as `conceptual metaphors’);
(b) The Philosopher’s hat: whose wearer’s responsibility is abstracting a coherent—albeit informal and not necessarily objective—holistic perspective of the external world from our sensory observations and their associated perceptions (corresponding to Rudolf Carnap’s explicandum, and to Gamez’s `C-theory’); and
(c) The Mathematician’s hat: whose wearer’s responsibility is providing the tools for adequately expressing such recordings and abstractions in a symbolic language of unambiguous communication (corresponding to Carnap’s explicatum; and to Gamez’s `P-description’ and `C-description’).
4. I would further view the above activities as providing merely the means by which an intelligence such as ours instinctively strives to realise its own creative potential within the evolutionary arrow of a, perpetually-changing, environment that not only gives birth to, but nurtures and encourages, a species to continually adapt to survive unforeseen and unforeseeable challenges.
5. From such a perspective, eliminating ambiguity in critical cases—such as communication between mechanical artefacts, or a putative communication between terrestrial and/or extra-terrestrial intelligences (whether mechanical or organic)—may be viewed as the very raison d’etre of mathematical activity. An activity which aspires:
(a) First, to the construction of mathematical languages that can symbolically express those of our abstract concepts—corresponding to Lakoff’s conceptual metaphors, and Carnap’s explicandum—which can be subjectively addressed unambiguously. Languages such as, for instance, the first-order Set Theory ZF, which can be well-defined formally but which have no constructively well-defined model that would admit evidence-based assignments of `truth’ values to set-theoretical propositions by a mechanical intelligence. By `subjectively address unambiguously’ I intend in this context that there is essentially a subjective acceptance of identity by me between:
(i) an abstract concept in my mind (corresponding to Lakoff and Nunez’s `conceptual metaphor’) that I intended to express symbolically in a language; and
(ii) the abstract concept created in my mind each time I subsequently attempt to understand the import of that symbolic expression (a process which can be viewed in engineering terms as analogous to my attempting to formalise the specifications, i.e., explicatum, of a proposed structure from a prototype; and which, by the `Sapir-Whorf Hypothesis’, then determines that my perception of the prototype is, to an extent, essentially rooted in the symbolic expression that I am attempting to interpret).
I would, however, qualify the (Sapir-Whorf) hypothesis of linguistic relativity as the assertion that language:
(iii) limits the expression of a thought (as distinct from the thought itself) to what can be expressed within the language by the vocabulary and grammar of the language; and,
(iv) influences how such an expression is understood under a subsequent interpretation of the expression, so that—even to the originator of the thought—that which is communicated in a subsequent interpretation of the expression need not necessarily reflect faithfully that which was sought to be expressed in the first place.
(b) Second, to study the ability of a mathematical language to objectively communicate the formal expression (corresponding to Carnap’s explicatum) of some such concepts categorically in a language such as, for instance, the first order Peano Arithmetic PA, which can not only be well-defined formally, but which has a finitary model that admits evidence-based assignments of `truth’ values to arithmetical propositions by a mechanical intelligence, and which is categorical, albeit with respect to algorithmic computability (see reference 1 below). By `objectively communicate categorically’ I intend in this context that there is essentially:
(i) first, an objective (i.e., on the basis of evidence-based reasoning in the sense of reference 1) acceptance of identity by another mind between:
* the abstract concept created in the other mind when first attempting to understand the import of what I have expressed symbolically in a language; and
* the abstract concept created in the other mind each time it subsequently attempts to understand the import of that symbolic expression (a process which can also be viewed in engineering terms as analogous to confirming that the formal specifications, i.e., explicatum, of a proposed structure do succeed in uniquely identifying the prototype, i.e., explicandum);
(ii) second, an objective acceptance of functional identity between abstract concepts that can be `objectively communicated categorically’ based on the evidence provided by a commonly accepted doctrine such as, for instance, the view that a simple functional language can be used for specifying evidence for propositions in a constructive logic.
Reference
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1. ‘The truth assignments that differentiate human reasoning from mechanistic reasoning: The evidence-based argument for Lucas’ Goedelian thesis’. In Cognitive Systems Research, Volume 40, December 2016, pp.35-45.
http://dx.doi.org/10.1016/j.cogsys.2016.02.004
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I always thought of mathematics as anything which could be reasoned completely precisely. If a concept can be described without ambiguity, then it is mathematical.
This is a very nice point of view, Shai, and it fits the stable marriage paper, and yet when mathematics meet reality some ambiguities arise.