What is Mathematics About?

 

 

What is Mathematics About?

 

Various suggestions have been made about this question: mathematics is about symbols, or mental constructions, or abstract Platonic entities. We can also ask what physics is about and expect a variety of answers: the sensations of the physicist, mind-independent material bodies, an all-pervading consciousness, abstract structure. In the physics case another answer has sometimes been contemplated: physics is about something whose nature we do not know and perhaps cannot know. This usually gets expressed as the thought that matter is an I-know-not-what, a mysterious substratum, a noumenal thingummy. We may know something of its structure and its mode of operation but we don’t know its inner nature. But I don’t know of any analogous view of the subject matter of mathematics: the view that mathematics is about something unknown to us yet partially described by our mathematical theories. Arithmetic, say, aptly represents the structure of mathematical reality, but nothing in it provides a clue about what numbers really are; nor do we have access to anything else that informs us of the nature of number. Thus we have agnostic realism about the mathematical world: numbers are real but we must be agnostic about the intrinsic character of numbers—as we must be agnostic about the true nature of what we call “matter”. Maybe physics and mathematics are ultimately about the same thing, but if so we are ignorant of what that thing is. The advantage of this way of thinking, in both areas, is that it allows us to avoid being forced into unpalatable positions: none of the standard positions is free of difficulty, and at least the agnostic realist position avoids these difficulties. Certainly the long history of mathematics gives the impression of people stabbing in the dark unaware of the vast mathematical world that would later be revealed; the very idea that mathematics has a subject matter would be alien to these early thinkers.[1] The reason is simply that we are not faced with any such subject matter by our senses or by anything else. There is a subject matter to mathematics, objectively real and determinate, but we have next to no knowledge of its ultimate nature; we don’t grasp the underlying mathematical reality (that very concept may be inadequate to its intended referent). The numerals we use are just symbols for we-know-not-what, mere placeholders. That, at any rate, sounds like an option to be added to the usual options. Call it mathematical agnosticism.           

[1] See for example Dirk. J. Struik, A Concise History of Mathematics (1987), which begins 10,000 years ago. Perhaps the earliest mathematicians would say that its subject matter consists of cows and corn, friends and foes.

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5 replies
  1. Giulio Katis
    Giulio Katis says:

    I was trained and worked briefly as a pure mathematician (post-doc researcher and lecturer), then moved to doing research with theoretical computer scientists, before transitioning to the dark side of financial markets/the trading floor environment (working in a very quantitative and data/technology intensive context). So you’ll have to excuse the nature and length of my reply.

    I was taught that there is a core (though not exclusive) driving force behind the development of mathematics, and this is to describe and understand the particular manifestations of the dialectic between space and quantity (to be understood very generally, which includes the dialectic of the discrete and the continuous, the concept of boundary, of transformation etc).

    For instance, as an undergraduate I was very taken (intoxicated one might say) with the abstract treatment of space, and that one could in an abstract way define notions such as dimension (via the concept/operation of boundary), as for instance done by Hurewicz in his seminal book Dimension Theory. But at a certain point, as an early post-grad, I received a wrap on my knuckles to not ignore quantity, and the relationship it has with space. (Otherwise I may as well just just become a logician, I was told). Geometry, for instance, involves moving back and forth between the logical relations of space, and the arithmetic and algebras of quantities. Perhaps the distinction “space vs quantity” may be in a sense artificial, both being different aspects of some underlying un-nameable reality (which the best our minds can come to terms with is via the dialectic of these two aspects).

    Obviously, a particular feature of mathematics, that has made it so powerful, is that it involves not just the development of concepts (and their logical relationships), but also that these can be expressed in various calculi. These calculi can be thought of as generalised algebras that allow one to combine, transform and operate on symbolic representations of spaces, quantities and, importantly, transformations of them.

    The mental process of calculation can encode a meaning, which must be borne in mind otherwise the calculations become blind/rote (which is sometimes the educational approach taken to train armies of engineers). This act of meaning-laden calculation, where the mathematician can shift from being the hedgehog to the eagle and back again, allows her to not just formulate proofs, but also explore and push the boundaries of her subject matter, potentially leading to new (useful) ideas or definitions. Of course, in reality we often just calculate blindly for a while, before pausing and seeing where it has taken us. Calculations of course allow (in typically a more formal context) the physicist and engineer to make models, predictions and designs – which can in turn lead to new mathematics.

    To take a closer look at what mathematics might be about, one might ask exactly what is the differential-integral calculus about? (I mean what is the calculus itself about, with its attendant concepts, operations, rules of transformation, and fundamental theorems. The answer is not the arithmetisation of analysis, which is how the subject is often taught, as if the content of the differential-integral calculus could be reduced to a specific arithmetic model of it, involving infinite series etc.) Specific physical laws and models require this calculus (it is the language in which they can be expressed), but the calculus itself is something that comes prior to these laws. It, and the rich and amazing generalisations of it developed in the 20C, presumably reflect something deep about the nature of the world (about the dialectic of space and quantity).

    Probably the two best known developments of the concept of space in the second half of the 20C, topos theory and non-commutative geometry (NCG), were both driven by problems requiring a better model of the tension between the continuous and the discrete, and reflect that a space can encode an active/dynamic element (contra the views that maths should take set-theory as foundational). For instance, rather than only recording how a part is passively included in the whole, one may also want to record how that part is the image of an active projection of the whole. What’s more, both topos theory and NCG codeveloped with the study and problems of algebras of quantities. Despite the commonalities just mentioned, they are very different types of mathematical theories, topos theory developed out of pure mathematics itself, while NCG was motivated by mathematical constructs that developed out of physics.

    In a much more basic way, calculi in general (and the process of calculation) allow mathematics to reflect something mysterious about the world: every non-trivial equality is not actually a strict identity. A very simple example: 5+3=8. The right hand side of the equation is a number, while the left hand side is an operation on a pair of numbers. As someone once said, equalities express transformation as well as identity.

    One might also ask what type of subject matter (in the natural sciences) would always remain beyond the domain of mathematics. Eg could mathematical concepts and calculi ever be developed to help understand the distinction between living and non-living? I have heard the view that this is unlikely, but I am not so sure. (Perhaps a future driving force for the development of mathematics, in its relation with the still nascent information sciences, will be the dialectic of state and process, feedback, inner reflections vs outer projections, data vs information etc.)

    I don’t know to what extent I have responded to your questions, but I would say that in addition to the question “what is mathematics about” one can also ask “what does mathematics allow our minds to actively participate in via meaning-laden calculations”.

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    • Colin McGinn
      Colin McGinn says:

      I enjoyed your reflections. They suggest to me the lack of clarity within mathematics about what the subject is about in the non-philosophical sense. But going beyond this there is still the metaphysical question about what category of being mathematical entities belong to–symbols, mental constructions, Platonic objects, or some other possibly inconceivable-by-us mode of being.

      Reply
      • Free Logic
        Free Logic says:

        I think that this entry and your response above point to the question about modes of being. Not a new question and not about to be resolved any time soon. In my opinion mathematical entities which are not in space/time, wherever/whatever else they are, support the view that there are different modes of being. “Modes are objective, logical, real: they are what they are and not some other thing.” McGinn, Logical Properties, p.83. As you said there though, you didn’t try to argue for this view in that book. Maybe it’s time?

        Reply
  2. Manuel Armenteros
    Manuel Armenteros says:

    Professor McGinn,

    I recently re-read Locke’s Essay and re-discovered his most excellent chapter, “Of the Extent of Human Knowledge”. It really is, as observed by Galen Strawson (and often quoted by Chomsky) one of the best chapters in the history of philosophy. To be precise, Strawson has one passage in mind, but I think it goes well beyond that.

    I’m sure you’ve read it. He is pretty much spot on in his observations about how little we know. Seems to me to be an unfairly ignored part of the book (comparatively speaking), which may be his best work in philosophy.

    Anyway, just wanted to share that.

    Very interesting essay as usual, which prompted this reply.

    All the best.

    Reply

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