The philosophy of mathematics -- what some people call the foundations of mathematics -- asks the big question ‘what is mathematics?’ -- and works through some of the classical and newer arguments in the field. Let us take a walk around in this territory and look for problems. At the end of this short visit, I will throw out a a few of my hunches – please feel free to do the same. This field is wide open.
What is mathematics? / a first look
Some traditional answers to the question ‘what is mathematics?’ are, that mathematics is about abstractions, mental constructions, symbols, hypothetico-deductive systems, or that it reduces to logic. None of these ideas ever entirely won the day, or overcame objections, or satisfied critics. The jury is still out, where it has been for the past several millennia. It has pretty much frozen in time where it stands today, ever since Gödel first published his Incompleteness theorem, in 1931.
To look at a map of mathematics -- as a quick wayfinder -- is to confirm this same impasse -- since different mapmakers map very different territories. Russell defined mathematics as the theory of implication, taking all mathematics into logic. Keith Devlin defines mathematics as the “science of pattern,” which again absorbs a part into a much larger whole. Rudy Rucker, Gödel’s student, organizes the field into five distinct categories: number, space, logic, infinity, and information. A glance at Wikipedia suggests the distinct parts quantity, structure, space, and change, but also a note to the effect that there is no generally accepted definition. Wolfram Alpha offers the definition: the science dealing with the logic of quantity, shape or arrangement. This seems to mush up logic and geometry.
Ben Orlin, Carl Sagan’s student, suggested several generic ideas: mathematics is creativity born from constraints; mathematics is the logic game of inventing logic games; mathematics is software/algorithms designed to perform a specific function; mathematics is somewhat like the caddy in golf: thus, not just a set of clubs, but knowing the right club for the shot. (Note: golf is the only game in which you assign yourself a penalty; this is like giving up an idea when you see that it is false).
For a while I experimented with this list: rules, counters, plug-intos, inside-ofs, and updates. This way of looking at the problem suggests that probability, and finding approximations for results, and the calculus, can all be considered together -- ultimately, they imply a temporal reference.
These are all speculations. There is no consensus about what mathematics is or even what it is about. Wittgenstein’s last entry in the Philosophical Investigations (Pt II, #14): “in psychology there are experimental methods and conceptual confusion” – also summarizes the situation that mathematics faces -- conceptual confusion and methods of proof. Geometry and analysis and statistics represent varying methods of proof, whereas the concepts underlying these methods remain confused -- e.g., space and time.
Wittgenstein regarded these questions as lying at the foundations of mathematics. He distinguished different kinds of explanation to help get at the subject. There are formal, mathematical, geometrical, and fully mechanical explanations. Somehow, we sense that a geometrical explanation unfolds more ‘reality’ than a formal one. A fully mechanical explanation shows how things work.
That is: some kinds of explanation are tautologies – what Kant called analytic truths – which merely restate definitions. Formalism – a grid structure – can sometimes “fit the facts” – at a certain juncture, it may not be possible to do more – at least we can stick with what we can prove – even if we can’t visualize anything.
Gödel showed that mathematics cannot be reduced to any mere formalism. Mathematics is therefore a way of knowing the world – a kind of insight or intuition – not merely the imposition of a grid. The problem is to capture what this is – how to think about this problem – by regarding mathematical axioms. Penrose says that “our guide in setting up procedural rules must be our intuitive understanding of what is self-evidently true simply given the meanings of the terms we use” (The Emperor’s New Mind, p. 110-1).
Hilbert expresses the same idea: “axioms express basic facts related to our intuitions of objects.”
If we can’t visualize anything, it is hard to see our understanding reaching very far – likewise a bare framework does not look like a good match for ‘reality.’
Einstein’s theory of gravity is stated in equations – but not merely equations – since one can picture (however imperfectly) curved spacetime – which makes this theory geometrical rather than merely mathematical: in this sense mathematics is “more than” mere formalism – if not yet a fully mechanical explanation.
Looking into this intuition more closely, e.g. in the case of Einstein’s theory, Wittgenstein’s comments ring true, because the mental images we conjure up to envision Minkowski spacetime, like rubber or plexiglass, or as embedded in higher dimensions, are all highly confused, even if they are better ideas than empty space. Spacetime is little better than its predecessor – the aether – except that the (now established non-existent) aether was a fully mechanical model (making use of what Newton considered “occult qualities” – “action at a distance”) where the newer (and well-confirmed) Einstein model is purely geometrical (gravity is curved spacetime).
Einstein’s own interpretation is that science is certain in the exact measure that it is mathematical. At the same time: “as far as the propositions of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality” (Geometry and Experience, 1921). This seems to push back towards a view about mathematics as logic. Einstein gradually adopts the view that we can capture reality via free creations of the mind, coordinated with dynamical variables, to run tests. We get at reality indirectly. Mathematics (including geometry) is a set of tools to help run the test. He is frank enough to confess that even the most abstract among the creations mind fashions to envision experience owe their existence from learning about the behavior of real objects – i.e., from intuition, as Hilbert and Gödel argued.
This view is fundamentally Kantian: reality is synthetic, i.e., both intuition and conception, therefore a construction, not a set of principles by which to guess the nature of reality. We get at reality indirectly, not then at the level of the concept, but at the level of the proposition, where we can test a sample hypothesis for its truth. Formalism is the ‘certainty’ part – methods of proof -- intuition is the ‘reality’ part – conceptual confusion.
Examples where mathematics stumbles into reality include Cantor’s discovery that there are actually more real numbers than rational; Russell discovery of a paradox in the very idea of set theory; Turing’s discovery that no Turing machine could ever solve the halting problem; and Gödel’s discovery, demonstrating the limits of formalized mathematical reasoning. Even with a big net, some truths will escape our net. As Austin said, “fact is richer than diction.”
In sum: mathematics is about procedural rules, but also about the world – part of mathematics is about order, part about discovery – minimally: mathematics is not one thing but plural. Mathematics is the toolkit but also knowing which tool to use when: thus a capacity of judgment. New tools always get added to the kit—e.g., regression to the mean. Carl Sagan cautioned that the toolkit can be "misued, applied out of context, or even employed as a rote alternative to thinking." "But applied judiciously, it can make all the difference in the world." "Skepticism boils down to the means to construct, and understand, a reasoned argument and – equally important – the ability to detect a fallacious one.”
What is mathematics? / some history
Pythagoras must have thought that the world was made of mathematics in some sense -- made of numbers -- numbers conceived as assemblies and ratios of units of magnitude -- which is why this whole way of thinking fell apart immediately after the Pythagoreans’ later discovery of irrational numbers, which cannot be expressed as whole number ratios.
Plato was drawn to the process of abstraction behind the conception of number: not five oranges or pencils or people, but the abstract notion ‘five’ these have in common. Plato originates the tradition of answering the question ‘what is mathematics?’ by talking about a special kind of object called a “form” (eidos) – an abstract object – as in the form of a triangle or a dodecahedron. This suggests the idea that a pattern or arrangement of parts is a kind of thing or reality in itself.
Thus among the things that exist are mathematical objects; these are abstract (non-spatiotemporal); and they exist independently of intelligent agents. This view is variously called Platonism or Mathematical Platonism or Object Realism. The basic ‘realist’ claim is that the mind discovers inhering structures in the universe -- ‘forms’ or ‘ideas’ -- rather than inventing them. Thus, we invent symbols -- which are arbitrary -- and by using them we discover realities -- which imply necessity -- we invent the language but (by this intermediary) discover the structure of reality.
Some of the things we discover to be real are material and concrete. Some are immaterial and abstract. Mathematics then is about the real world but more narrowly about the most abstract elements of it. Mathematics is the science of pure thought.
Plato introduces the term chorismos (separation) to help identify what he is talking about. He says that the way we learn mathematical principles shows us their purely noetic quality -- they are separate -- they cannot be sensed but are intelligible -- they are seen and comprehended by being demonstrated -- and so because they are the main examples of what we can learn about the world when we try to demonstrate exactly what we know, they are not just ta pragmata, things insofar as we have to do with them practically, but ta mathemata, things insofar as we are able to learn about them -- mathematical things -- things that we are able to demonstrate and to teach others exactly as we have learned. This implies a way of getting at reality not via the senses but by demonstration – thus an idea of reality more extensive than what the senses reveal – also an aspect of a reality apart – separated.
Aristotle’s view is that abstract properties reside in the objects from which they are abstracted. This means that mathematics is not about something chorismos or separate.
Aristotle is not disputing the import or truth of results in mathematics, but the nature of mathematical thinking, “so that our controversy is not about their being but about their mode” (Metaphysics 1076 a 36). In his dispute with his teacher Plato, he argues that mathematical truths are not separate from but are dependent on human beings. Therefore a number, e.g. three, has no independent nature, but is just a way of talking about something being “so many,” as for example so many trees (Metaphysics 1080 a 15). In the Physics he expresses this idea by saying that “to be a number is to be some number of given things” (221 b 14).
Aristotle’s criticism of the chorismos thesis implies that -- since they are separate -- abstract objects would have to exist prior to sensible things, but he thinks it makes much more sense to think that sensible things exist before anything can be addressed and counted and thus have a “number” (Metaphysics 1077 a 15).
Aristotle thinks Plato is making the same mistake in jumping from five oranges and five people to the abstract idea ‘five’ as he makes when he jumps from ‘man’ to ‘men’ to ‘Man.’ Aristotle is claiming: There is no idea of Man separate from human beings -- this is just a confusing way of talking about properties of the biological species ‘man.’ This is just a way a talking and does not imply any metaphysics -- it’s just semantics – poor use of language.
‘Mathematics’ then is just shorthand for various discoverable properties in the real world. Thus, beginning from Plato’s ‘realism,’ Aristotle offers the first ‘nominalist’ correction. Let’s not mistake the obvious.
This is the background for Aristotle’s discussions about first principles in the Organon and in the first chapters of the Physics, where he discusses what we should consider real and that reality cannot be one (undivided) in the sense in which Parmenides argued (185a 20).
According to tradition, Euclid’s discussion of first principles in the first book of the Elements continues Aristotle’s line of thinking in the Organon. What Aristotle calls “hypotheses of existence” Euclid divides into definitions (horoi), postulates
(aitêmata), and common notions (koinai ennoiai). In a sense, Euclid is collecting together in one place everything that people had learned about mathematics in all the ancient schools -- the Pythagoreans, the Eleatics, the Academy, the Lyceum. Euclid develops the classic example of developing a structure deduced from principles, so much so that for medieval thinkers like John of Salisbury, Walter Burleigh and William of Occam, mathematics simply is the Euclidean demonstrative method from basic axioms.
What impressed Euclid’s more distant successors was not his demonstrative system but his building process itself -- thinking of mathematics very narrowly as the process Euclid undertakes e.g. in his first proposition: Book I Proposition 1 -- “to construct an equilateral triangle from a given line segment.” Euclid does not say whether or not a triangle exists; he begins talking about points and lines as a matter of course and applies his postulates until the thing is done (hoper edei poiesai, Latin ‘quod fieri’) or until the statement is finally proved (hoper edei deixai, Latin quod erat demonstratum). Thinkers like Brouwer and Heyting developed so-called “intuitionist” mathematics from Euclid’s process of gradually assembling an object of study without explicitly calling attention to doing this. In Euclid’s words, mathematics is a kind of poetry -- poiesai, ‘to make,’ and deixai, ‘to show,’ ‘to bring to light’) -- mathematics brings a thing about and makes use of it -- thus e.g. to show geometrically how to build an equilateral triangle or a geodesic dome -- a view called ‘constructivism’ today.
After the classical age, the great names in mathematics emerge in the Islamic world -- Al-Khwarizmi, Alhazen, Omar Khayyam -- from whom we get zero, negative numbers, letters to symbolize quantities of different kinds, the use of the unknown, the method of balancing and rebalancing equations via cancelling terms across an equality, and the general strategy of reduction to simplest terms, which came to be known by Al Khwarizmi’s name -- reasoning algorithmically -- thus laying out the first principles of algebra (Ar. al-jabr, ‘reunion of what is broken’).
Russell once explained in a beautiful phrase that “in algebra the mind is first taught to consider general truths” and that the entire point of learning mathematics is to hone exactly this ability to deal with strictly abstract truths. Alhazen makes the jump to experimental reasoning and Fibonacci introduces the Hindu-Arabic number system to the West. Al Khwarizmi, Alhazen, and Fibonacci lay the foundations of the scientific method.
This brief history -- antedating the modern world -- motivates mathematical realism, nominalism, logicism and constructivism – still the foundational ideas in the subject today. This demonstrates that the philosophy of mathematics is dominated by ideas from its early history, long before the revolutionary work begins in the Enlightenment. It is literally stuck in the past.
What is mathematics? / the second crisis
The new language of algebra made it possible to collect all mathematical knowledge within a single language -- what Descartes calls ‘analytic geometry’ -- a language made entirely of abstract expressions. But then this form itself opens up as an object of study -- i.e., thinking began to abstract from abstractions. Thus, in a sense, Euclid abstracts from experience in conceiving the idea of a mathematical point, but Cardano (1545) is simply working through equations when he is forced to take the square root of a negative number. Bombelli (1572) seizes on this idea as the imaginary number, and the complex number line extended into 2D, as a way of conceiving vectors in the imaginary plane.
The history of mathematics is driven by problems -- in response, thinking creates new tools -- new tools then inaugurate a fresh round of thinking about foundations.
This lets us catch a glimpse at the relation between science and philosophy.
Science cannot really get off the ground without an account of what it is doing -- a philosophy. Philosophy cannot make progress without laying down some foothold in knowledge -- a science. Philosophy and science -- the search for knowledge and knowledge itself -- evolve together. Mathematics is both science and philosophy – thinking and thinking about thinking. Thinking about thinking in terms of the growth of mathematical learning demonstrates how open-ended the thought process really is. Thinking about thinking gets us to the insight that there is no final vocabulary of thought. We may still discover radically new tools and therefore begin to ask radically new questions -- to be driven back to rethink the foundations once more – in sum, a caution against dogmatism. Change is real – if only the change in our own understanding.
Leibniz fills notebooks with approaches for calculating the area under a curve -- in 1675 he develops a fresh approach with some new definitions -- he defines the idea of a constant; of a variable; of a dependent variable and an independent variable; and the idea of the function that relates them. He defines the derivative of a function as the rate of change of a function with regard to its independent variable. Therefore, a function y of a variable x, like y = f (x), takes various values given various inputs, which Leibniz conceives as ordered pairs (x,y) of points on a Cartesian plane -- later as triplets (x,y,z) on a 3D grid – later still reasoning about infinities and even possible worlds beyond merely actual ones.
He defines the derivative of a function f (x) = dy / dx -- the change of y over the change of x -- and defines the differential as any such arbitrary change in the value of x, thus computing the differential dy = f (x) dx. Extrapolating, he conceives the differential as an infinitesimally small change in value, which he expresses mathematically as the symbol d as we do today. He conceives the integral, symbolized with the elongated s symbol as we do today, as the sum of a sequence of subdivisions, i.e. of differentials, as the anti-derivative, an idea known today as the fundamental theorem of the calculus.
The area under a curve is the integral of the ‘slices’ under it to the x-axis:
A = f (x) dx
Newton, who also discovered these “principles of calculus” independently, immediately saw that they have physical meaning. If we think of a point moving in space as a variable y coordinate along an extending x-axis, the rate of change of the area bounded by the curve is this very curve itself. Thus, the tangent to the curve at any arbitrary point is its instantaneous rate of change -- an infinitesimal increment of time -- the derivative f (x) is the slope of the tangent to the curve y = f (x) at any arbitrary point on the abscissa.
Newton models the physical universe as made up of bodies each having some form of attraction to one another, as natural weight or mass or heaviness (in Latin gravitas), as the earth attracts the moon, the sun the earth, or an apple falling to the ground. By observation, neither the moon nor the apple moves in a straight line at constant speed. Therefore, an attractive force exerts an acceleration on a distant body (an important form of which is the acceleration of falling objects near the surface of a body due to gravity).
Velocity is the rate of change of distance with respect to time -- the derivative of the distance function (rate times time equals distance) -- acceleration is the rate of change of velocity with respect to time -- the second derivative of the distance function. With these ideas, we have constructed mathematical tools for describing what Newton calls “the system of the world” interrelating any object in the world, the forces acting upon it, and its motions.
F = 0 ↔ dv/dt = 0 [law of inertia]
F = m dv/dt = ma [impulse is (J = F dt)]
Fa = - Fb [to every action an opposite and equal reaction]
An object at rest will stay at rest unless a force acts upon it -- an object in motion will not change its velocity unless some force acts upon it. Force is the derivative of momentum over time -- momentum is mass x velocity -- force is mass x acceleration. These are what Newton calls the “mathematical principles of natural philosophy” -- Philosophiae Naturalis Principia Mathematica -- published in 1686.
The Principia defines philosophy as “arguing from phenomena to investigate the forces of nature.” Newton says that he is cultivating (excolo) mathematics in so far as it relates to philosophy, which implies that mathematics is something we have to work on, to nurture and try to improve. Mathematics is not a ‘thing’ -- it is more like a task or a set of tasks – a thing (or things) one is obliged to create and nurture -- something we dream up on the spot as we confront a specific kind of problem.
Newton remarks that his discovery of universal gravitation demonstrates that things that seemed to have nothing to do with one another -- falling objects and the motion of the planets -- were forms of one principle. Magnetism and electricity were later understood to be forms of one thing -- electromagnetic (EM) waves -- light was later recognized to be an undulation of EM fields -- space and time were seen to be aspects of the same continuum -- spacetime -- ST-- then gravity was seen to be a kind of curvature of ST -- EM fields were seen to coincide with the weak force – thus electroweak interactions – EW -- this process continues with theories of quantum electrodynamics and quantum gravity. Thus, from the initial problem of finding roots for algebraic equations – a problem inherited from Al Khwarizmi -- Leibniz and Newton ended up inventing an entirely new language for understanding for understanding change – i.e., for everything that exists.
What is mathematics? / theory and practice
Newton’s principles were quickly applied to astronomy, mechanics, civil engineering, metallurgy, chemistry -- shipbuilding, the design of cannons -- bridges and clocks. This raises the following fascinating question – how comes it that this method has been so spectacularly successful? The method appears to arise almost out of nowhere simply through trying to respect the basic rules for maintaining algebraic closure over basic operations.
How can we explain the amazing match between pure theory and amazing practical applications?
The tradition of modern German philosophy, beginning with Leibniz and Kant, and carried on by writers like Husserl, Weber, Jaspers, Heidegger and Hannah Arendt, make this the central question in the philosophy of mathematics. Arendt concludes her studies with the admission that the greatest perplexity in the entire state of affairs we are looking at today is that completely fictional and arbitrary ideas can and do describe actual physical processes and thus can be seen to “work.” She says: it’s as if our methods outpace our understanding -- therefore mathematical truth may not even be comprehensible to human reason. And so with Newton we have a second, major puzzle in the philosophy of mathematics -- something on the order of Pythagoras’ terrifying discovery of irrational numbers. (E. Wigner, 1959 -- “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communication on Pure and Applied Mathematics 13 (1960), 1-14).
Kant thinks he knows the answer to this question. He thinks that he has learned it from Galileo. Galileo says famously that the laws of nature are written in the language of mathematics. Kant agrees and says further that there is only so much science in a thing -- that is, knowledge of a thing -- as there is mathematics in it. But Kant says that he is making a Copernican turn -- reversing the basic premises of understanding. Thus, he says it is not that the laws of nature are written in the language of mathematics, but that we ourselves must bring a new template for understanding reality by thinking explicitly in mathematics.
“Reason has insight only into that which it produces after its own design” (KdRV B xiii).
Kant’s new idea is that mathematical statements are a priori synthetic judgments. That is: mathematics is a priori, rather than a posteriori -- it is ‘before’ (it is prior, i.e. it must be assumed beforehand) rather than ‘after’ (meaning what occurs in experience; it is not learned in experience). This ‘before’ structure becomes the main subject of thinking about mathematics for centuries afterwards -- trying to understand mathematics in relation to the consciousness-making machinery in our minds. The oddity here is that we do not have any access to the consciousness-making machinery in our minds -- only to their results.
Husserl uses many different terms to talk about the ‘before’ structure -- “forerunner ideas” -- the “already-made world” -- the “bases from which” we develop every concept we have. He looks for a way to talk about the primordial pre-reflective laying out of the world, which he conceives as a kind of embedded power coeval with the advent of language.
Part of his idea here is that a linguistic creature is a mathematical creature -- these are two ways of looking at the same problem – we are looking at the same indirect approach, the same synthetic construction.
In a way, it seems natural that our methods outpace our understanding, because we take in much more than we are able to make sense of.
Husserl emphasizes that the output of creating a structure in a symbolism (e.g. by facial expressions, movements of the hands, sounds, grunts, signs and scripts), immediately becomes independent of the objects the structure was intended to model. Thus, a brand-new development begins as a result of translation into the new medium. This ‘transformation’ is the thing Husserl is trying to understand.
This is the natural criterion for any kind semantics or theory of invariant meaning.
This is what Husserl considers as the key to identity across platforms – i.e. to error-free translation. Husserl’s student Martin Heidegger calls symbolic transformation “the fundamental presupposition for knowledge” (What is a Thing?)
“Ta mathemata, mathematical things, are things insofar as we take cognizance of them as what we already know them to be in advance. Thus, we already know in advance what counts as a physical object, we already know the plant-like character of plants, we already have an idea of what counts as an animal and the animal-like characteristics that alert us that what we are looking at is an animal. Thus, the kind of learning that we are talking about -- ta mathemata, things insofar as they can be learned --- is an extremely peculiar kind of seeing and taking in which we are taking something we already have.”
The mathematical project anticipates the essence of things -- it defines the advance blueprint for the structure of things in relation to everything possible -- it is the orienting guide which at the same time is the fundamental measure for laying out the universe. As Wittgenstein remarks in On Certainty: we use judgments as principles of judgment. There is no hard-and-fast distinction between axioms and theorems.
Heidegger explains further that the mathematical method of getting at a thing -- the mode of access appropriate to axiomatically pre-determined basic planning -- works via the cycle of hypothesis-experiment-output, designed to posit conditions in advance and then, by applying reason and, gradually, by using instrumentation, posing key questions that precise observation alone can answer.
For this reason, he claims that the mathematical project “skips over” things in order to look for facts. This is ultimately Kant’s achievement -- the achievement of the Enlightenment -- the spirit of experimental natural philosophy – the ability to skip over the immediate outlook to look for underlying principles.
Thus, we see that if the question is, why does this work?, we quickly see that in many cases it doesn’t -- we try something and it doesn’t work out -- what we are looking at here is precisely a cycle of experiments and results.
What is mathematics? / mathematics as action
In a sense, Kant offers the precedent for getting away from thinking about mathematics as a body of knowledge -- he wants us to consider the proposition that mathematics is something that human beings do. There is a verb buried somewhere in mathematical thinking -- a function or an action -- this is what Heidegger means by “skipping over.”
But then even this formulation is misleading since it invites us to make an important mistake, that of essentializing into a single basic ability or faculty what is in reality an enormously complicated array of techniques that are completely different from one another. The attempt to force them into one model – e.g. via Langlands “sheaths” – works against this grain.
The word ‘mathematics’ is plural in most languages, which suggests that whatever else mathematics may be, it is a collection. So, a next step in reasoning is to accept that mathematics is a big collection -- more like a toolbox than a language -- which must include things like counting, measuring, abstracting, demonstrating, applying techniques like differentiation or integration or graphing, or taking an average.
Going with the toolkit metaphor, let’s examine this more closely. The reason we are reaching into the bag is to do something -- to solve a problem -- so we are looking around for a tool -- we have a bunch of tools, but we need the right one for the job.
So to begin with there is the world, and we are going to use mathematics to solve a problem in the world, which means that mathematics is something we bring with us to experience -- we carry it with us and make use of it when we need to -- which is why Plato thinks of mathematics in terms of anamnesis, or recollection -- calling a thought to mind -- just as Aristotle thinks of mathematics in terms of the most basic categories by which we orient ourselves in life -- mathematical ideas are forerunner ideas -- or overall a set of tools by which we take the measure of a thing, examine it – especially to ask questions.
In a sense what we are talking about here is modeling -- we are trying to come up with a mathematical model for something that is happening in the world -- and so the point of talking about the toolkit is to emphasize that there is always more than one way to do this.
Note that in mathematics we often switch around between techniques as we dig deeper into a problem and confront new issues. So, e.g. you are evaluating an integral, and there is some odd expression in the argument. So, you look for a simplification strategy -- this could be a dummy variable, or factoring, or completing a square, or evaluating by parts, or using a trig substitution, or coming back at the end to look at the geometry. The action of switching back and forth between spatial intuition and numerical competence -- between seeing and counting – is one kind of checking strategy -- there are also other pieces like remembering a useful identity or approximating a sum or pulling out an expression to look at later.
Looking more closely at the idea of a mathematical model, it jumps out that a mathematical model is a different thing than the thing itself that it is meant to help see. One of the big issues in the philosophy of mathematics is simply to grasp this point -- not to mistake our strategy for getting at a thing as if it were the thing itself.
Consider an example -- consider an equilateral triangle T standing upright with vertex a at the top, with vertices denoted <a,b,c>. Now rotate the triangle counterclockwise from the top around its midpoint.
The triangle has three upright positions with a vertex pointing upwards: the beginning point 0, then rotated 2 over 3, and then rotated again 4 over 3 radians. Now if we consider this simple scenario as a situation in the world, we could say that in this situation S, in the observable universe, there are three physical states possible -- modeled as various abstract states Q. Immediately we see that we can model these three states in the following different ways:
Set Q1 = ( <abc>, <cab>, <bca>)
Set Q2 = (0c, 2 / 3c, 4 / 3c)
Set Q3 = (1, 2, 3)
The point is that the identical situation in the universe can be managed by different modeling assumptions. Note that the abstract states could be a number but might not be a number. From the standpoint of nature, there cannot be any preferred description between Q1, 2, 3 ... but in some cases one kind of Q may make more sense or be more helpful to use … practicality, pragmatism, approximation become important – making use of what you have. It's as if the strategy is roughly go ahead and adopt this convention, and then see where you get.
Probably Cantor meant something like this when he said that the essence of mathematics is freedom (Ben Orlin offers another definition of mathematics: “mathematics sees itself as the auteur director of an experimental art project”).
In mathematics, one is able to imagine any situation and then try to get at it by making different kinds of assumptions. This is why mathematics is so freeing.
There is no constraint from reality -- we simply construct a tool and see what it does. Sometimes you look in the kit and reach for the wrong tool -- whatever you try doesn’t work -- but this is not a problem in itself but is good information -- this is the important thesis that Imre Lakatos put forward in the 70s – to get real – to see that it’s unhelpful to expect too much -- to have too highfalutin’ an idea about what we are doing -- mathematics is simply heuristics -- what we’ve been taught is wrong -- mathematics is not about eternal truths – mathematics is about improvisation -- trying out ideas and seeing what works.
It's somewhat like the idea of thinking about the number 54, which is 54 in base 10. But 54 is 120 in base 6 and 2000 in base 3 and 110110 in base 2.
Is it important to note that the name, convention, system, or handhold we use to grab onto the thing we are looking at is our own creation?
Yes. This helps to keep us humble. We are basically just playing with our toys. It also tries to get us back to the (puzzling) fact that we do not have access to the reality-making mechanism at work in our own consciousness -- only to its results – only to the results we notice and investigate.
By studying these problems, we begin to see that if we fiddle with the inputs, we get slightly different outputs, which forces us to stare down how ‘human’ the real world really is.
This is roughly what Frank Ramsey was getting at in his ideas about “partial belief” (thought conceived as a guessing-game). This breaks down the standard of absolute certainty and begins re-standardizing in favor of ordinary human judgment
-- the human world is not a logical system. Nor is there any likelihood of forming a consistent description of all the phenomena we are looking at by the choice of one metaphor over all others – let’s create an alternative structure: probability.
This is somewhat like the situation of the electron. Is the electron a particle, or is the electron a wave? The electron behaves like a particle when measured moving in an external EM field. The electron behaves like a wave if it is diffracted by a crystal. This suggests a perspectival or aspectival or activist aspect of knowledge – as Kant saw -- what exists has to do with which questions we are asking – what we look into and discover – without which we have no knowledge at all. Thinking is a synthesis, an amalgam, not a single overall algorithm.
This result is sometimes referred to as the Grothendieck principle: there are no absolute beginnings; parameters always determine a class of objects, not just an ‘object’ per se. There are settings and re-settings, default vs. client-altered base routines – maps and new maps. Topology stays abreast of changes in flow – phase transitions motivate re-mapping. The theory of invariants should be subsumed under the theory of representations. In effect invariants are temporarily flat variables. Maxima, minima – the sector for ‘existence’ is roughly a fuzzy intersection.
In effect: we’re used to thinking that information about an object — say, that a glass is half-full — is somehow contained within that object. The lesson emerging from these studies is – roughly – that this is not the case -- objects do not exist like this – objects do not have an independent existence with definite properties of their own -- instead, objects exist only in relation to other objects.
What is mathematics? / the third crisis
Continuing to dig deeper into these questions, it is helpful to build a model such as we are discussing and then see what can be learned – to choose something to look at in the world and try to build up a system as a wireframe for examining it.
To take a subject – what about a mathematical model of love? -- roughly a problem in dynamics, or in systems that evolve over time – perhaps a system of simultaneous differential equations. Briefly, with a few assumptions, one can make a start in modeling the environment in which people fall in love, which could be frictionless or resisting; modeling mate choice, romantic styles, the complexity of emotional reactions, influences on mood, standards of general sociability, ideas about romantic outcomes -- also the chances of divorce vs. long-term relationships. One could easily amass a few results and arguably these might indeed have some predictive power -- they might even be applied in real-time scenarios on the model of a control system. Knowledge like this could perhaps help a person estimate risks and make better decisions in his or her love life. It could make one more rational. But it would be odd to claim that, because of such calculations, anyone understood love any better.
What is most obvious here is that there is nothing in the mathematics of love that directs how this knowledge should be used -- evidently mathematical tools can be used for good or ill. Plato sees this truth -- for him it is the principal result to focus on in thinking about thinking. Alexander Grothendieck, probably the greatest mathematician in the last century or so, who died in 2014, also emphasized this idea -- i.e. the deep problem in mathematics is the moral problem -- i.e. what to do with it.
In physics there is something called “the Copenhagen interpretation,” which is something that Neils Bohr and Werner Heisenberg created in the 1920s to help people interpret all the new data being gathered about quantum processes and how to go about controlling them. This interpretation was accepted at the time and is widely accepted today as the working model for understanding interactions at the smallest scales we can measure. Yet the Copenhagen interpretation remains fundamentally problematic in that it cannot explain or reconcile the particle-wave duality -- it basically just gives up on this question. So, we have a useful scientific theory that allows us to make predictions that are confirmed by experiments, but one that is philosophically unsatisfying -- more accurately, it is philosophically incoherent. We have a method that works but no real understanding of what it means. This returns us to Wittgenstein’s pronouncements.
By common consensus the most important single result in contemporary mathematics is Gödel’s discovery of the incompleteness principle. Gödel noted (as we have above) that the entire history of mathematics takes the position that mathematics is something a priori -- it represents the basic instruction-set or blueprint for the universe -- and therefore is something that we ourselves bring to experience, as a basic condition, before we actually ‘have’ any experience at all. Thus, as Wittgenstein said, there can be no surprises in mathematics. But Gödel’s discoveries did come as a surprise! That is the intrusion of reality. It is surprising to learn that a mathematical proposition might be true even though there can be no possible way of proving it.
Put another way: human beings can know truths which they cannot prove -- no system of axioms is capable of generating all the truths of mathematics -- which Gödel interpreted to mean that we have a kind of intuition for truths that sweeps beyond the reach of any logical system. Ultimately mathematics is about this power rather than any of its creations.
Mathematics is everything that results from this primordial, improvisational, risk-taking, language-making, world-making intuition-sense-detector-grid power.
Gödel’s thesis precipitated the third crisis in the foundations of mathematics -- the crisis of incompleteness -- inspiring innumerable ‘rescue’ attempts to unify mathematics on new principles -- just as Pythagoras’ problems with irrationals and Leibniz’s problems with infinity drove the earlier histories of mathematics -- from which we get e.g. Bombelli’s imaginary numbers and Cantor’s set theory. Recent attempts include Category theory from the 1940s, Grothendieck’s Scheme theory from the 1960s, and the Langlands program from the 1970s, still ongoing.
For the present, there is no consensus about the subject -- mankind is where Gödel left it.
Still, in thinking about Gödel’s achievement, we bump into some terrific ideas about how to think about mathematics – and pretty much everything else -- as a big collection – as a toolkit – decidedly not as the ‘master plan. ‘
Let us call the toolkit the toolkit – the Tk for short. Gödel shows that we cannot fit everything we are doing into one category – therefore into the Tk. (In a way it is equally misleading to suggest that everything can fit into a fuzzy category). In any case, the Tk is bigger than all our attempts to systematize it. Somehow the Tk stretches beyond whatever the Tk has managed to create (“cultivate”) up to now.
What is mathematics? / a few hunches
Mathematics is a tool -- a set of tools -- a toolkit – the Tk -- also experience making use of the tools and seeing how things turn out. Therefore, the clubs, the caddy proffering which club to use in which situation, and also some experience in caddying, and the gradual development of know-how in this métier -- since this is not the sort of thing one can learn in a day: mathematics is, in a word, toolkit savvy.
In his Art of Discovery, from 1685, after having invented the calculus, Leibniz expressed the hope that one day a truly universal form of calculation might replace opinion and thus put an end to all disagreement:
“The only way to rectify our reasonings is to make them as tangible as those of the mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can say: Let us simply calculate, to see who is right.”
Great as he was, what Leibniz says here represents a fundamental mistake in thinking about the subject. Disagreements are as fundamental to mathematics as are computing sums. Disagreeing is another kind of mathematical tool -- another ‘data collection device’ -- another way of sending out signals and getting feedback as they return. The point is that we cannot replace judgment with mere calculation.
Leibniz wandered one day through the marketplace in Hanover and wondered to himself how to conceive the skill on display everywhere around him. He recognized that there is no account – or at least none yet – of ordinary expertise based on a set of underlying rules: “The most important observations and turns of skill in all sorts of trades and vocations are as yet unwritten” (1680). He imagines being able to replace judgment with calculation but is forced to concede that -- up to a certain point -- the thing we call skill or expertise or intelligence might not even be theoretical, i.e., rule-based, or algorithmic, or (in mathematical terms) recursive. Judgment might be intuitive or holistic or perceptual in a way that belies algebraic formulization.
If it were possible to formalize all skills in mathematical terms, then a mathematics of mathematical skill would be exactly the sort of thing which Gödel deconstructs. He proves that a mathematics of mathematics is impossible. This is why Gödel talks about mathematical intuition.
We cannot endow any system with everything we have learned, or with all our good sense and experience, and thus be done with human judgment. To try to do this would be like trying to replace wisdom with a kind of machine -- which would be insane -- which is why it is so striking that so much of human history has been devoted to doing just this – which is why the peculiar focus of the present fascination with producing an artificial intelligence seems so wrong-headed.
An important result here is that thinking about mathematics from a metamathematical standpoint -- the study of mathematics using mathematical methods -- such as is attempted in Frege’s Begriffsschrift (Concept-script, 1879), and in Russell and Whitehead’s Principia Mathematica (1910), and projects from later times such as the Langlands program (beginning 1967) -- a proposed “unified field theory of mathematics” – is of absolutely use to us. Even if this work were successful, it could not help us. The point is judgment, irreducible to any recipe, judgment that would preclude any of us from ceding our own power of judgment to a machine.
Reality is distinct from strict predictability – despite the "Einstein-Podolsky-Rosen criterion of reality," which states that "If, without in any way disturbing a system, we can predict the value of a physical quantity in that system with certainty, then there exists an element of reality corresponding to that quantity” – here again Einstein disagrees with Einstein. Einstein anticipates so much, including modeling for us the import of disagreement – he signs on with one group of scientists to test a consequence of the QM model (e.g., entanglement), but also signs on with another group trying to disprove the very same thesis (therefore to disprove “spooky action at a distance”). Einstein lays the foundation for quantum mechanics -- he also rejects he very rules he laid down to create the field – because he keeps asking new questions, trying out new schemes, considering his own ideas incomplete, still not seeing the hidden variables – always returning to the first issue: what is really there?
In sum: there is no general answer to the question, what is mathematics?, as there is no general solution to the problem of solving problems -- there can be no Copenhagen solution in philosophy -- no Idiot’s Guide to problem-solving -- we can’t get ourselves out of this, or do an end-around the moral issue of what to work on and why.
The Tk keeps changing -- the Tk is what we make of it – the Tk keeps evolving – because we go on, we face new situations. The problem is to take up the Tk as an ethical project -- to accept responsibility for it without presuming that we know exactly what it is -- the problem is to go on cultivating mathematics as a form of philosophical questioning. / 2019
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Updates to the Tk: nudges, boosts, the 'black swan' principle, optimal noise, the 'broken leg' principle in decision hygiene, confirmation bias, anchoring, the disposition effect, the "psychological immune system," and bullshit receptivity – the tendency not to think. Frankfurt wondered whether the growth in knowledge was matched by an equal growth in ignorance. The sphere of knowledge keeps growing – is this also true about ignorance? / 2024
