Some philosophers think mathematics exists in a mysterious other realm. They’re wrong. Look around: you can see it
Aeon Magazine
The mathematical worldJames Franklin | Professor of Mathematics at the University of New South Wales in Sydney. His book An Aristotelian Realist Philosophy of Mathematics is out this month.
Here is the publisher's blurb:
An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge.
See our debate over Aristotelian realism in the comments.
Aristotelian realism is now being put forward in a number of fields, I think erroneously in that it has little to do with Aristotle's sophisticated explanation of intellectual intuition for his time that modern thought has rejected. I see the contemporary version of "Aristotelian realism" as being simply a reassertion of naive realism that ignores the epistemological issues that have raged since ancient times and have yet to be resolved in a way that is either logically compelling through philosophical reasoning, or on the basis of basis of scientific evidence.
Why is this important for economics? Because it is the basis for the assertion of "laws" that are supposedly logically necessary (a priori) and self-evident as principles underlying the structure of reality, hence, universal, absolute, and ahistorical. It is essentialist and structuralist thinking in contrast to contextual and constructivist. Critics also view it as dogmatic rather than scientific.
17 comments:
Franklin's work is very good. His "Aristotelian realism" about mathematics is not a wholesale re-introduction of Aristotle, but a more limited application of some Aristotelian ideas to the more limited problem of understanding the role of mathematical concepts in natural science.
You might find this interesting:
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.pdf
Traces the implications of combining a Platonic view of at least some parts of math (the natural numbers and basic arithmetic) with the assumption that our minds are some form of digital machine.
Dan, my objection is calling something "Aristotelian" realism that his not based on either Aristotle's epistemological theory of intellectual intuition that accounts for how the mind is able to know what is "real" independently of it. This is the basic issue.
Why bring in Aristotle at all. Why not Locke's realism. Or common-sense realism that appeals to self-evidence and simply rejects Hume.
There is a whole history to this debate and if someone claims to be in the Aristotelian tradition then one has to either take over Aristotle's explanation as correct. This has usually been done by interpreting it as Neo-Thomists Jacques Maritain and Etienne Gilson did.
My impression is that this kind of contemporary Aristotelianism has about as much relation to Aristotle as Ayn Rand and Murray Rothbard, who also claim to be in the Aristotelian tradition.
If one chooses to be a realist, then unless one chooses to be a naive realist, it's incumbent to make a case for how human beings are able to overcome dualism the apparent dualism of mind and material world and resolve Hume's fork.
Aristotle was not a naive realist. Neither was Locke. They were addressing different issues. Aristotle was reacting to Plato's theory of forms, and Locke was writing post Descartes in a scientific era in which the mechanics of perception were beginning to be understood.
The context today is one of cognitive psychology and consciousness studies.
But the basic issues remain controversial.
I left out the "or" in the "either-or" I set up.
my objection is calling something "Aristotelian" realism that his not based on either Aristotle's epistemological theory of intellectual intuition that accounts for how the mind is able to know what is "real" independently of it, or some reinterpretation of Aristotle's view consonant with modern knowledge.
Ken, the problem with finding the basis of math in the mind is accounting for how it applies to nature. Many idealists ("Platonists") and Kantians attempted to do this by saying that knowledge structures the given in accordance with its own structure. The difference between idealists and Kantians is that idealists think that mind actually structures reality, which is why reality reflects our mental structures, while Kantians think that mind imposes its structure in the knowing process, so that we do not actually know things in themselves. Realists of all stripes reject both these types of view.
Let me simply it further at the risk of oversimplification. The fundament epistemological positions are monistic or dualistic, which are ontological distinctions.
There are two monisms. The first, comprising idealisms, is that everything is mental and therefore reality is reflected in the mind through knowledge so that mind is knowing itself. The second, comprising materialisms, is that everything is material and the structure of matter is therefore also the structure of mind, and knowledge is mechanical. Notice the the metaphor for reality in idealism is mind and the metaphor in materialism is a machine or an organism.
Dualisms hold that mind and matter (world) are separate and distinct, so the problem of whether an how knowledge bridges the gap arises.
There are two other ways of approaching the problem that avoid the epistemological problem. The first is pragmatic and holds that the monism v. dualism debate is useless, so let's just move one and forget about it as being a pseudo-problem.
The other way is nominalism. Nominalisms holds that meaning is constructed, e.g., from contextual usage. Necessity and contradiction are syntactical. How logical and mathematical constructs compare with the world is probabilistic . "Reality" is constructed and stochastic.
There are advantages and disadvantages involved in all these types of explanation, and what one faction regards as a feature, another faction sees as a bug.
I personally think at least some of math exists independently of any mind or any physical universe.
You should give that paper a read if you have time ... he shows a reversal, with mind deriving from math itself, rather than the other way round.
Yes, then the question becomes where does the math come from as pre-existent. The ancient answer was the world of forms as in Plato and other views that posit a causal world), or as divine ideas in the mind of God as in Augustine.
Those answers are no longer accepted in modern circles, so the challenge then becomes providing a contermporary explanation for that which pre-exists mind and matter.
I think he's on fairly solid terminological ground Tom. The account is Aristotelian in the sense that he is proposing a middle way between nominalist reductions of mathematical universals and Platonists accounts. On Frankin's view, mathematical universals and structures are "in" the objects that exemplify them, rather than occupying some sort of ideal realm of which the objects are just shadows. That kind of immanent realism is typically associated with Aristotle and some of his medieval successors.
I don't think "Lockean" would capture the spirit of what he is doing, because it has nothing to do with Locke's way of ideas. And Locke had a very undeveloped philosophy of mathematics.
His view is not a version of naive realism. That is, as far as I can tell, he is not arguing that we have some kind of unmediated acquaintance with the objects of perception. It's a view about mathematical ontology, not epistemology.
Here's the paper of which I assume Franklin's book is a more comprehensive elaboration:
http://web.maths.unsw.edu.au/~jim/irv.pdf
The account is Aristotelian in the sense that he is proposing a middle way between nominalist reductions of mathematical universals and Platonists accounts. On Frankin's view, mathematical universals and structures are "in" the objects that exemplify them, rather than occupying some sort of ideal realm of which the objects are just shadows.
If they are "essences" or "structures" *In the world* of matter, how do they get into the mind, which is non-material? In other words, it presumes realism. Aristotle explained it with his theory of intellectual intuition.
What I am saying is calling it "Aristotelian" doesn't advance the debate without some sort of Aristotelian explanation.
What is here the assertion of real structures, and now in sociology the assertion of "real causal powers" but no explanation that accounts for how they are known to be "real."
I don't think that such people see the fundamental issues that have been debated for millennia, and are just asserting naive realism and thinking it is some sort of advance.
Let me be clearer in more detail for those unfamiliar with ancient philosophy.
Plato held that the intelligible forms that objects in the sensible world "participate in" exist independently in a the causal world of intelligible form which is knowable by intelligence (nous). It is well-known that Socrates was attracted to Anaxagoras's proposal that nous is fundamental but he was disappointed that Anaxagoras only used it as a launching pad for a mechanistic philosophy. (Plato, Phaedo 97b8ff.) However, it seems clear from the Cave Analogy that the forms exist for Plato's Socrates in a supersensible world that is supremely intelligible in that it is illumined by the "sun." The sun here is a symbol of intelligence. These forms are the patterns of things and when we see the patterns in things we "remember" the forms which are already in our deep memory.
Aristotle rejected Plato's narrative, metaphorical, and dialectical method for a categorical one that resemble the reasoning with which we are still familiar. In this sense, Aristotle established philosophical method subsequently. So claiming connection with Aristotle has to be more specific than that.
Aristotle correctly saw that Plato was actually talking about causes but indirectly using metaphor. Aristotle developed a theory of causality based on Plato's metaphors. The sensible world and the objects that constitute it have four causes (aitia),
1. the material cause which is unknowable since matter (hyle) is unintelligible,
2. the formal cause or essence (ousia), which is intelligible and which the active intellect acquires directly through intellectual intuition in the process of knowing. The active intellect impresses the form of the object known on the passible intellect as "the blank tablet" of the mind that is capable of receiving intelligible. This are the Platonic forms acting as the formal cause of things in things. This is the basis of Aristotle's realism.
3. the efficient cause that actually effects change, and
4. the final cause, or "end" (telos) in the sense of purpose. This is Aristotle's adaptation of Plato's the Good that attracts everything to itself.
Point to note: Aristotle asserted than we know the forms of things directly through intellectual intuition and explained the process through the dual faculty of the active and passive intellect. This is not a process of abstraction in which the mind acts on the given and constructs concepts. Universal concepts are known directly as the forms of the things and are imported into the mind through intuition in a similar fashion to the sensible qualities of things being known through sense intuition that gives knowledge of particulars.
The Constitution of Man
The constitution of man, as considered in the following pages, is basically threefold, as follows:
I. The Monad, or pure Spirit, the Father in Heaven.
This aspect reflects the three aspects of the Godhead:
Will or Power - The Father
Love-wisdom - The Son
Active Intelligence - The Holy Spirit
and is only contacted at the final initiations, when man is nearing the end of his journey and is perfected. The Monad reflects itself again in
II. The Ego, Higher Self, or Individuality.
This aspect is potentially
Spiritual Will - Atma
Intuition - Buddhi, Love-wisdom, the Christ principle
Higher or abstract Mind - Higher Manas
The Ego begins to make its power felt in advanced men, and increasingly on the Probationary Path until by the third initiation the control of the lower self by the higher is perfected, and the highest aspect begins to make its energy felt.
The Ego reflects itself in
III. The Personality, or lower self, physical plane man.
This aspect is also threefold
A mental body lower - manas
An emotional body - astral body
A physical body - the dense physical and the etheric body
The aim of evolution is therefore to bring man to the realization of the Egoic aspect and to bring the lower nature under its control. [DK]
So, any consciousness standing before the 'SUN' where both spirit and matter merge in one energy - would see his 'spark' the Monad, the first differentiation of this one energy, as separate to the consciousness that views it: but in full knowledge that consciousness, spark and Sun are one - a ray descending from this 'individualised' spark to establish consciousness as the Ego in the soul body, from which in turn descends a ray to establish consciousness as the 'I' in the persona. The distance between the Monad and the 'I' greater than between the stars and the streetlamps, and the affairs of the 'I' far below, of comparable significance. The return path is evolution and its door is within the human heart, because that is where the Life aspect of the descending ray is anchored. Mind on this path becomes the 'eye' of the Ego in the lower worlds, not something on which to paint thoughts (speculation) and emotion.
Of course we think the 'I' is very important! It is what makes the world go around.
Thanks for the link, Dan. I skimmed the paper (it's long and tightly argued). Many interesting points and he recognizes that the basic issue is getting the real into the mind as it is in the real. I don't think he is successful in bridging that gap although he basically incorporates intellectual intuition as insight without getting into how it works, as Aristotle does in detail.
He gives reasons to believe the real is captured by the mind, but no cognitive or epistemological explanation. Lacking that, I think that claims like following will be unconvincing to opponents although it might bolster those with confirmation bias.
Here are a few quotes:
"An essential theme of the Aristotelian viewpoint is that the truths of mathematics, being about universals and their relations, should be both necessary and about reality. Aristotelianism thus stands opposed to Einstein’s classic dictum, ‘As far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.’ [Einstein, 1954, 233]. It is clear that by ‘certain’ Einstein meant ‘necessary’, and philosophers of recent times have mostly agreed with him that there cannot be mathematical truths that are at once necessary and about reality."
***********
"Certain schools of philosophy have thought there can be no necessary truths that are genuinely about reality, so that any necessary truth must be vacuous. ‘There can be no necessary connections between distinct existences.’
"Answer: The philosophy of mathematics has enough to do dealing with mathematics, without taking upon itself the refutation of outmoded metaphysical dogmas. Mathematics must be appreciated on its own terms, and wider metaphysical theories adjusted to take account of whatever is found."
Or, thinking from the top down:
The reality is that there is ONE, functioning on all planes as a point of its energy, manifesting as planned activity by means of a path of Light. I realise that reads like gobbledegook or scares a lot of people, but that is like being scared of your shadow, in reverse! People actually know this - especially when the tsunami is rolling in or the last breath rolls out. Why forget when the sun is shining?
By the time you get down into human activity it is dreamtime (conjecture about what is seen in front of the eyes; aka mind). The 'reality' we create for ourselves. There is the ‘god’ that we create for ourselves whom we house in the mind: - then there is that universal Energy out of which everything is made. In a democracy we vote for mind; cling to it like a life-raft! Philosophy goes around and around in the vortex, looking for a loophole. Psychology is caught up in the 'I' which in this world is either one man's jetsam or another man's flotsam. Death comes and draws a line through them all, which is around about the time most people realise that they haven't got a clue about anything - least of all their existence! It shouldn't be like that! Human beings should know what is inside of them. What is inside of a human being is actually very very beautiful, peaceful too!!
Maths belongs in Time and Space and that is what it is good for. There is a place in Consciousness, where time and space stay on the shore! All there is, is Being; as far as the eye can see. And it is beautiful; absolutely tranquil, ultimately powerful.
It's a funny place to visit planet earth? Then again we have almost poisoned the place completely and do nothing but argue all day long.
It's true that modern Aristotelian realism (e.g. David Armstrong's work on universals) avoids epistemological issues. In my book, An Aristotelian Realist Philosophy of Mathematics) there are three chapters on epistemology, which I hope successfully connect perception with an Aristotelian-like theory of intellectual intuition.
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