Is Non-Conceptual Content a Kind of Given?

We can go back to the 18th century and to Thomas Reid to see what appears to be a reference to what is now called “non-conceptual content”. Reid wrote:

“[S]ensation, taken by itself, implies neither the conception nor the belief of any external object. [] Perception [on the other hand] implies an immediate conviction and belief of something external.”

Now let's jump to the first half of the 20^th century, when “sense-data theory” was in fashion. An experience of a sense-datum was seen in non-conceptual terms. It is of course true that this position was problematic in many ways. As Michael Williams puts it:

“It is important to see that acquaintance with sense-data is 'direct' or 'immediate' in two senses. Not only is it independent of any further beliefs, it is pre-conceptual. It makes propositional knowledge (involving conceptualisation) possible.”

Now we can jump forward to another distinction offered by Laurence BonJour. He argues that what he calls “sensory content” is not in fact “propositional or conceptual in character”.

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If we return to the sense-datum of a red patch.

That must have meant (i.e., to sense-data theorists) that a red patch sense-datum is firstly experienced and only then perceived as a red patch. That is, the concepts [red] and [patch] are applied to the red patch. (What is meant by “applied” here?)

Thus, the sense-datum of a red patch is a basis of later knowledge, not an example of knowledge itself. It can become conceptual. So there's no reason to believe it is incorrigible, direct, immediate, pure or anything else like that. The Given is only seen as the Given after the fact.

Following on from that, this raises the possibility (or actuality) that even if we accept non-conceptual content, we may still go wrong when we describe it. However, how would we (or anyone else) ever know that? Here we have another problem: the problem of private mental states. (In this case, private sensory content.)

As for BonJour's position.

We can say that such “sensory content” can become conceptual content. And only then can it (as Michael Williams puts it) “justify basic beliefs”. More simply: sensations cause beliefs (i.e., they aren't themselves beliefs). This is a position which will be defended later.

This question must now be asked:

What is the precise nature of the movement from non-conceptual content to conceptual content?

Animals

It can be argued that some animals have the same “non-conceptual content” as human persons – at least in certain cases. It's just that we can (or do) apply words or concepts to that very same content. This is a position expressed (if not endorsed) by Alex Byrne when he states the following:

“Some of the perceptual states of lower animals have contents in common with human perceptual states.”

We can of course ask (in tune with philosophers like Daniel Dennett) how we could ever know (or even surmise) that “animals have contents in common with human perceptual states”. On the basis of their behaviour? (How would that work?) On the basis of their evolutionary proximity to human persons?

In any case, Martin Davies also allows animals non-conceptual content when he says that “human infants and certain other creatures” are “arguably [ ] not deployers of concepts at all”. More specifically, this perceptual content is free “of any judgement that might be made”. 

We can also take the position of Brian Loar.

Let's talk about a dog called Fred.

According to Loar (not his own example), Fred “picks out a kind” - the kind human beings call “human beings”. More correctly, Fred picks out a particular (say, Mike) and sees that he/it belongs to the kind we call (though it doesn’t) human beings. At no stage of the game is Fred’s x is F anything like our x is F. That is, it's not linguistic or sentential.

So how close must Fred’s x is F to our x is F? More relevantly, why do we demand an exact parallel with our x is F in order to allow attributions of concepts and beliefs to dogs and other animals?

Having put the dog's position, it doesn't appear to be an argument for non-conceptual content in that Loar says that Fred (the dog) “picks out a kind”, etc. Clearly, noting or picking out kinds (even if non-linguistically, etc.) doesn't seem like a candidate for non-conceptual content.

Nonetheless, something (or some things) must also come before the linguistic expressions of human persons: both as a species and as individuals. Our linguistic expressions didn’t occur ex nihilo.

Here’s Paul Churchland making related points:

“[L]anguage use is something that is learned, by a brain already capable of vigorous cognitive activity; language use is acquired as only one among a great variety of learned manipulative skills; and it is mastered by a brain that evolution has shaped for a great many functions, language using being only the very latest and perhaps the least of them. Against the background of these facts, language use appears as an extremely peripheral activity, as a species-specific mode of social interaction which is mastered thanks to the versatility and power of a more basic mode of activity. Why accept, then, a theory of cognitive activity that models its elements on the elements of human language?”

Peacocke, Davies and Tye

One direct case of purportedly non-conceptual content is offered up by Christopher Peacocke when he writes the following:

“Only those with the concept of a sphere can have an experience as of a sphere in front of them."

In detail:

“The natural solution to this... quandary is to acknowledge that there is such a thing as having an experience of something as being pyramid shaped that does not involve already having the concept of being pyramid shaped.”

This is problematic in that there may still be concepts involved in this experience. That is, it may still be what Peacocke calls “object-involving”. In any case, this person (or animal) has an experience - and even an experience of a something (x). It's just that he (or it) doesn't have an experience “as of a sphere”. It's a sphere to us; though not to him (or it).

Elsewhere, Peacocke says that the

“content of experience is to be distinguished from the content of a judgement caused by the experience”.

In our example, this would be a judgement that x is a pyramid.

In basic terms, the “content of experience” and the“judgement” don't occur at precisely the same time. Whether this temporal way of looking at things makes sense (or is acceptable) is another matter. I say that because it can argued that the experience and judgement occur at one and the same time.

Peacocke also sets up a relation between the non-conceptual and the conceptual when he says that

“thought can scrutinise and evaluate the relations between non-conceptual and conceptual contents and obtain a comprehensive view of both”.

This question must again be asked here:

What is the precise nature of the movement from non-conceptual content to conceptual content?

Colin McGinn points out the position represented by philosophers like Peacocke when it comes to “representational content”. McGinn says that they accept

“prerepresentational yet intrinsic level of description of experiences: that is, a level of description that is phenomenal yet noncontentful".

Peacocke himself says that “sensational properties do not determine representational content”. (Peacocke cites an example of an array of dots which can be seen as either vertical or horizontal rows^{1 and} Wittgenstein's duck-rabbit seems to be relevant too.) Concepts are part of this story. 

More specifically on representation, Peacocke says that the

“representational content of a perceptual experience has to be given by a proposition, or set of propositions, which specifies the way the experience represents the world to be”.

Does Peacocke mean something linguistic or something abstract here? Does he mean that these propositions need to be articulated or verbalised? And why do we need propositions at all in these cases?

Peacocke also appears to make a mistake about a supposedly non-conceptual experience.

He says that a person “waking up in an unfamiliar position or place [will have] minimal representational content” . Yet surely unfamiliarity doesn’t entail lack of conceptual or “representational content”. This person may still conceptualise his “unfamiliar position or place”. Perhaps this only displays Peacocke’s linguistic or propositional bias. This person may not describe or form “propositional judgements” about his unfamiliar position or the new place he finds himself in. However, that may be irrelevant to conceptual and/or representational content.

To summerise Peacocke's position:

sensation ⟶ perception ⟶ judgement

Martin Davies too makes an explicit distinction between what he calls the “perceptual content of experience” and the having (or possession) of concepts. In detail:

“[T]he preconceptual content of an experience is a kind of non-conceptual content. What this means is that a subject can have an experience with a certain perceptual content without possessing the concepts that would be used in specifying the content of that experience.”

Thus, in Davies's jargon, the “perceptual content of experience” comes first, and only then are concepts applied to it. Indeed, it seems possible that this perceptual content of experience may remain (as it were) free of concepts.

Davies's perceptual-content idea is even more radical in that it “is not object-involving”. The basic point here seems to be that non-conceptual content can't (by definition?) be object-involving.

Martin Davies also divorces perceptual content from the “representational”. That is, the non-conceptual implies (to me at least) the non-representational. According to Davies, that non-conceptual “substrate” is added to by a “representational superstructure”. In addition, the “sensational” is also distinguished from the “representational”.

Finally, Michael Tye also makes a temporal division when he says that

“visual sensations feed into the conceptual system, without themselves being a part of that system”.

One can now ask how “visual sensations” can “feed into the conceptual system” without having some kind of vital connection to that system. And if they have such a connection, then can they still seen as non-conceptual? That is (as asked twice before), what is the precise connection (or link) between the non-conceptual (visual sensations in this case) and the conceptual system? One is tempted to infer (as with John McDowell) that in order for these visual sensations to be able to feed into the conceptual system, then they must share something with that system. Otherwise how does that system distinguish irrelevant sensations from relevant ones?

Despite that, according to Tye, “phenomenal content” is by definition non-conceptual. And he too (like Peacocke) gives us his own division between “phenomenal content” and belief. Thus we have:

“A content is classified as phenomenal only if it is nonconceptual and poised [for use by the cognitive centres].”

And:

“Beliefs…[which] lie within the conceptual arena, rather than providing inputs to it.”

Here “phenomenal content” is seen as input to be worked upon later. Beliefs, on the other hand, are outputs. Is this phenomenal content “the Given”? It certainly seems to be when Tye made the epistemic point that phenomenal content is “poised for use by the cognitive centres”. That is, it comes (epistemically) before such “use”.

We also have an explicit tying of concepts to language from Tye when he writes the following:

“Having the concept F requires, on some accounts, having the ability to use the linguistic term 'F' correctly.”

Yet Tye also cites mental content that is non-linguistic when he says that

“after-images, like other perceptual sensations, are not themselves thoughts or beliefs; and they certainly do not demand a public language”.

It's very hard to see afterimages as conceptual. Then again, afterimages are a very special case of mental content. That is, afterimages don't seem to be relevant to this discussion because they're unlikely (or rarely) to be the basis of later conceptual content.

Critics of Non-Conceptual Content/the Given

Wilfrid Sellars

Wilfrid Sellars (in his 'Epistemic Principles') lays his critical cards on the table when he said the Given

“would be a level of cognition unmediated by concepts; indeed it would be the very source of concepts”.

Sellars was right to imply here that there can't be a “level of cognition” which is “unmediated by concepts”. However, we needn't also conclude (or accept) that these sensations can't be a “source of concepts”. The problem is that Sellars fuses cognition with non-conceptual content. Yet the two needn't go together. Indeed not even an old-fashioned believer in the Given would have believed that. And isn't that why non-conceptual content is a (to use Sellars' own word) “source” of cognition and concepts, not an example of these things?

Incidentally, Sellars himself did make a distinction between the two when he said that a “sensory element [of perceptual experience] is in no way a form of thinking”. So Sellars happily concedes a “sensory element”. Having said that, that sensory element may not come first. It may simply be part of the perceptual experience from the very beginning.

John McDowell

John McDowell explicitly believed that the acceptance of non-conceptual content is effectively a rebirth of “the myth of the given”. McDowell is now well-known (i.e., within epistemology) for holding this position. That is, for his rejection of the idea that we have a temporal division (as I see it) between the mind being presented with a non-conceptual Given, and then a later application of concepts to that Given.

McDowell himself writes (in his Mind and World):

“[T]he content of a perceptual experience is already conceptual. A judgement of experience does not introduce a new kind of content, but simply endorses the conceptual content, or some of it, that is already possessed by the experience on which it is grounded.”

What does McDowell mean by “simply endorse the conceptual content” (i.e., apart from claiming that the content is already conceptual)? What does McDowell mean by “already conceptual” here? How does a concept-based “judgement” differ from the already conceptual “content of a perceptual experience”? And isn't all this dependent on what philosophers take concepts to be?

So McDowell believes that we're fooled into believing that there's such a thing as non-conceptual content. In the case of “rich” experiences (such as when we don't have the words or concepts for particular shades of colour), it's nonetheless the case that we have a “a recognitional capacity, possibly quite short-lived, that sets in with the experience”. (This is similar to the stress which David Lewis made on “recognitional capacites” when discussing the what-Mary-didn't-know scenario. That is, Mary doesn't "acquire any new facts". However, she does acquire new recognitional abilities.)

Despite all that, it's hard to see a situation in which having a “recognitional capacity” can ever be mistaken for a non-conceptual state by so-called “non-conceptualists”. After all, McDowell argues that it's fully conceptual. So perhaps McDowell is wrong to assume that this is an example of a conceptual state mistaken (by non-conceptualists) for a non-conceptual state. In other words, don't non-conceptualists have something more basic in mind when they refer to non-conceptual content?

A Rich Experience

It's often said that an experience which is “rich” can't be entirely conceptual. For example, Alex Byrne quotes a fictional person saying:

“It appears to me that my environment is thus-and-so.”

And who then says:

“So I suppose the content of my experience is rich/perspectival/phenomenal/non-conceptual.”

I'm not sure if this is a good way of putting it because if the environment is seen to be "thus-and-so”, then doesn't that imply that it's not “non-conceptual”?

Anyway. The above is a simple way of saying that this person had an experience of a particular environment which he couldn't completely describe. Or, rather, at the actual time of the experience he didn't describe it; though afterwards he may well have been able to do so. Nonetheless, even after the experience, there would still have been elements of that environment for which he has no words or concepts.

So does this show that this experience was at least partly non-conceptual?

Here's another description of a rich experience from Christopher Peacocke:

“Our perceptual experience is always of a more determinate character than our observational concepts which we might use in characterising it. A normal person does not, and possibly could not, have observational concepts of every possible shade of colour...”

Can we say that an experience is “more determinate” even if it's not conceptualised in any way? In what sense, then, is it determinate? Here there's a hint at a kind of discrimination which doesn't involves concepts, let alone words or descriptions.

Despite that, even if a person has no “observational concepts” or words for “every shade of colour”, that person is said to still note the unnamed shades of colour. But how is that possible without concepts? Perhaps the problem is tying concepts too closely to public words. Surely an animal (say, a dog) can discriminate without public words or “observational concepts”.

Bill Brewer articulates this point in the following:

“For surely a person can discriminate more shades of red in visual perception, say, than he has concepts of such shades, like 'scarlet', for example.”

That seems to be the case. However, according to Brewer, the “conceptualist” has an answer to this. It is to

“exploit the availability of demonstrative concepts of color shades, like 'that_r shade', said or thought while attending to a particular sample, R”.

As can be seen, this is still language-fixated in that although there are no public words for these colour shades, this person is still saying “that_r shade” to either himself or to another person. Surely this rules out any discriminations an animal may make.

So here we may have a “fineness of grain” (or “richness”) without concepts or judgements that implies a level of discrimination which occurs without concepts - or at least without words.

Of course one can apply concepts or words after the fact. One can even invent one's own neologisms for experiences or colours one doesn't know the name of. But we may still have had an experience of the colours without using concepts or words.

We can now go beyond talk of the different shades of colour and say, as Gareth Evans did, that perception itself always (or often) has a “phenomenological richness” which goes beyond the concepts used in perception. In other words, experiences or perceptions are more fine-grained than can be accounted for by simple references to the many different shades of colour. Indeed, phenomenologically, an experience is almost infinitely rich (or detailed). And even if we had public words for everything within it, such words would still never be used during the actual experience itself. 

Davidson on Causes and Sensations

It may be useful to press-gang Donald Davidson into this debate.

Whereas we can stress non-conceptual content, Donald Davidson himself stressed the “causes” of what he called “sensations”. So, in this picture, the causes of sensations can be said to take the place of non-conceptual content.

In the following passage (from his paper 'A Coherence Theory of Truth and Knowledge') Davidson wrote:

“The relation between a sensation and a belief cannot be logical. Since sensations are not beliefs or other propositional attitudes. What then is the relation? The answer is, I think, obvious: the relation is causal. Sensations cause some beliefs and in this sense are the basis or ground of those beliefs. But a causal explanation of belief does not show how or why the belief is justified.”

So it's worthwhile rewriting this passage for clarification and to put it within the context of non-conceptual content. Thus:

The relation between “sensations” (or non-conceptual content) and beliefs cannot be logical. Since sensations are not beliefs. What then is the relation? The answer is, I think, obvious: the relation is causal. The world causes non-conceptual content (or sensations) and in this sense is the basis or ground of concepts (or conceptual content) and beliefs. But a causal explanation of non-conceptual content (or sensation) doesn't show how or why judgements about it are true or false of the world.

Another passage (from the same paper by Davidson) is even more apposite in this context. Davidson wrote:

“Accordingly, I suggest that we give up the idea that meaning or knowledge is grounded on something that counts as an ultimate source of evidence. No doubt meaning and knowledge depend on experience, and experience ultimately on sensation. But this is the 'depend' of causality, not of evidence or justification.”

Here again we can rewrite Davidson in the context of our take on conceptual and non-conceptual content. Thus:

I suggest that we give up the idea that conceptual content (or belief) is grounded on something that counts as an ultimate reality: either non-conceptual content or the world. No doubt conceptual content (or beliefs) ultimately depend on non-non-conceptual content (sensations) or the world. But this is the 'depend' of causality, not mind-independent truth or fact.

In Davidson's terms, conceptual content causally “depends” on non-conceptual content (or sensations) and the world (or causes). However, that non-conceptual content or world doesn't - and can't - in and of itself guarantee us truth. Thus it can't be seen as the Given in the 20th century sense.

In that case, perhaps we can say that our causal interactions with x are the Given; though not the beliefs these causes bring about. In that case, as Susan Haak says, it is

“only propositions, not events [or objects], that can stand in logical relations to other propositions [or beliefs]”.

The causal interactions (or non-conceptual content/sensations) themselves are neither beliefs nor propositions. Therefore they can't “stand in logical relations to other propositions” or beliefs. The causal interactions (or sensations) are causes of beliefs; though, in and of itself, they are neither evidence for such beliefs nor justifications for further beliefs.

In any case, the same causal context - taken only in itself - can cause different beliefs in different people and possibly different beliefs in the same person at different times. The interpretations of our causal contacts depend on our prior beliefs and the prior concepts which we apply to our causal interactions. And even if a particular causal contact brings about the formulation of new beliefs or new concepts, these will still be dependent upon - or be related to - prior beliefs and prior concepts.

Conclusion

To sum up. It can be said that surely there must be some kind of Given in order to get the ball rolling.

Thus in Davidson's scheme we had:

causes ⟶ sensations ⟶ beliefs 

instead of the more basic:

sensations (or non-conceptual content) ⟶ beliefs (or conceptual content)

Alternatively:

i) An experience of x.

ii) Then an experience of a [?].

Or:

i) Sense experience x.

ii) Then sense experience x + conceptual content (or plain concepts).

We can now say that i) and ii) may not, or cannot, occur at one and the same time.

Thus “the Given” needn't remain given. That is, i) becomes ii).

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Note:

1) 

Christopher Peacocke's example of a "sensational substratum" of either three vertical columns or four horizontal rows doesn't seem to work. He says that "we see the array as three columns of dots rather than as four rows". However, isn't that simply because the dots in the columns are closer together than the dots in the rows? If the distances between the dots were identical in both cases, then what would we "see"? The image at the top of the page is more balanced than Peacocke's own example in this note.

Max Tegmark, Our Universe is Not Mathematical

The first thing to say is that the claim that “the universe is mathematical” hardly make sense at a prima facie level. It's not even that it's true or false. So this must surely mean that it's all about how we interpret such a claim.

Despite saying that, sometimes it's hard to express (or even understand) precisely what Max Tegmark's actual position is. Can we say that reality (or the world) is mathematics or mathematical (as in the “is of identity”)? That reality is made up of numbers or equations? That reality instantiates maths, numbers or equations? Or should we settle for Tegmark's own very radical words? -

“The Mathematical Universe Hypothesis... at the bottom level, reality is a mathematical structure, so its parts have no intrinsic properties at all! In other words, the Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks.” 

[See later comments on Carlos Rovelli, Lee Smolin and relational theory.]

To put the case formally and as clearly as possible: Tegmark believes that physical “existence” and mathematical existence are “one and the same” (which is a phrase he often uses) – they equal one another. More specifically, Tegmark stresses “structures”. Thus if we have a mathematical structure, it must exist physically as well. Or, more strongly, all mathematical structures exist physically.

The Unreasonable Effectiveness of Mathematics

Tegmark mentions Eugene Wigner a couple of times in his book and he's clearly inspired by his well-known question.

Wigner once wrote:

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

As for the “miracle” of mathematics and its applications:

“It is difficult to avoid the impression that a miracle confronts us here, quite comparable... to the two miracles of laws of nature and of the human mind's capacity to divine them.”

And then we have the question which Tegmark quotes a couple of times in his book:

“The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and... there is no rational explanation for it.”

Albert Einstein also asked the same question in the following:

“How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”

However, Einstein's conclusion appears to be radically at odds with both Wigner's and Tegmark's:

“[...] In my opinion the answer to this question is, briefly, this: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

So Does the “unreasonable effectiveness” of electricity or roads also “demand an explanation”? That ironic question is asked because there are indeed explanations of maths effectiveness. However, I feel that they won't satisfy Max Tegmark.

The Mathematical Universe Hypothesis

The drift of Tegmark's central position is that the only way we can justify our belief in a mind-independent reality is to accept that it is mathematical. As it stands, of course, that almost seems like a non sequitur. Tegmark does provide some argument for this position; though very little. Indeed only a small part of his book (Our Mathematical Universe: My Quest For the Ultimate Nature of Reality) is devoted to this central thesis.

Perhaps Tegmark is not alone in his position. Take this very Tegmarkian utterance from the physicist Brian Greene in which he deprecates what Tegmark calls “baggage”:

"The deepest description of the universe should not require concepts whose meaning relies on human experience or interpretation. Reality transcends our existence and so shouldn't, in any fundamental way, depend on ideas of our making."

Interestingly enough, Tegmark's juxtaposition of mathematical realism with metaphysical realism was preempted by Hilary Putnam in 1975. Putnam spoke of Wigner's “two miracles” (i.e., the power of mathematical descriptions of the world and the mind's “capacity to divine them”). Putnam concluded that in order to be a metaphysical realist and a believer in mind-independence, one must see the world as mathematical. (This is part of Putnam's “indispensibility argument”, which is not strictly platonic in any way.)

Tegmark's position on the mind-independence of reality is different to most positions advanced by metaphysical realists. Reality is not mind-independent in the metaphysical realist's sense. It's independent of human beings (or minds) simply because it's an abstract mathematical structure. Thus this has little to do with whether reality is observed; the way it's observed; its verification; etc. Reality is independent of human beings even if (or when) humans observe it.

So let me sum up that in a basic argument:

i) Mathematics is mind-independent.

ii) All non-mathematical descriptions of reality are mind-dependent.

iii) Therefore in order to achieve a true mind-independent description of reality, one must only use mathematics (or mathematical structures) to do so.

One part of Tegmark's argument is that if a mathematical structure is identical (or “equivalent”) to the physical structure it “models”, then they're one and the same thing. Thus if that's the case (that structure x and structure y are identical), then it makes little sense to say that x “models” - or is “isomorphic” with - y. That is, x can't model y if x and y are one and the same thing.

Tegmark applies what he deems to be true about the isomorphism of two mathematical structures to the isomorphism between a mathematical structure and a physical structure. He gives an explicit example:

electric-field strength = a mathematical structure

Or in Tegmark's words:

“' [If] [t]his electricity-field strength here in physical space corresponds to this number in the mathematical structure for example, then our external physical reality meets the definition of being a mathematical structure – indeed, that same mathematical structure.”

In any case, if x (a mathematical structure) and y (a physical structure) are one and the same thing, then one needs to know how they can have any kind of relation to one another. This truism displays this problem:

(x = y) ⊃ (x = x) & (y = y)

In terms of Leibniz's law, that must also mean that everything true of x must also be true of y. But can we observe, taste, kick, etc. mathematical structures?

In addition, can't two structures be identical (if not numerically identical) and yet separate?

All this is perhaps not the case when it comes to mathematical structures being compared to other mathematical structures (rather than something physical). Yet if the physical structure is a mathematical structure, then that qualification doesn't seem to work either.

All this is also problematic in the sense that if we use mathematics to describe the world, and maths and the world are the same, then we're essentially either using maths to describe maths or the world to describe the world.

In addition, Tegmark's Mathematical Universe Hypothesis (MUH) is certainly played down by Israel Gelfand when he writes:

“There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.” 

In the end, Tegmark is telling us what physicists have always believed about the importance of mathematics when it comes to describing the world. However, he's adding the perhaps unjustified conclusion that both “structures” are one and the same thing.

Platonism/Pythagoreanism

It's easy to see that Tegmark's position is pythagorean or platonist in that it stresses mathematical entities. However, it is platonist (at the least) in a much stronger sense in that it states that only mathematical objects or structures exist. (Whether Plato held that position is for others to decide.) As a consequence of this, Tegmark's position must also be a form of monism in that literally everything is deemed to be mathematical. (Just as former monists believed that reality - or the world - was all “spirit”, matter, “neutral” or God.)

Tegmark is at his most platonic and radical in the following passage:

“... a complete description [of external physical reality] must be devoid of any human baggage. This means that it must contain no concepts at all! In other words, it must be a purely mathematical theory, with no explanations or 'postulates'...”

Yet how can one even construct a single thought without “no concepts at all”?

Thus when Tegmark talks about mathematics, it's clear that he's stressing that it is in no way “human”. That it's not what he calls “baggage”. Yet even if mathematics is an abstract Platonic realm, it is still human beings that gain access to it. It's still human beings that give such abstract objects and equations names or symbols. It's still human beings that make use of these abstract entities. And it's still human beings who may be getting it all wrong.

In terms of detail, we can ask if Tegmark is correct to say that time and space are “purely physical objects”. He says the same about “curvature” and particles.

Thus when Tegmark says that a dodecahdron was never created, he's saying what Plato might have said. Similarly, when he says that dodecahedron doesn't exist in space or time at all, that too is pure Plato. To quote the New Scientist:

“A dodecahdron was never created, says Max Tegmark of the Massachusets Institute of Technology. A dodecahedron does not exist in space or time at all - it exists independently of both. Space and time themselves are contained within large mathematical structures, he adds. These structures just exist; they cannot be created or destroyed.”

Of course Tegmark adds to Plato when he also argues that such things as “space and time themselves are contained within large mathematical structures”. The purely platonic statements are easy to grasp (at least because they've been on the table for two thousand years); though Tegmark's addition of space and time to the Platonic world is (as it were) harder to grasp.

Tegmark also says that “the rectangular shape of this book [doesn't] count” when it comes to “geometrical patterns such as circles and triangles” and their being “mathematical”. So why don't “human-made designs” count? In addition, is it really the case that the “trajectory” which results from our “throwing a pebble [and] the beautiful shape [a parabola] that nature [then] makes” is more precise, symmetrical and exact than anything we human beings can construct, as Tegmark claims?

This reminds me of problem the mathematician Richard W. Hamming notes when he writes the following:

“We select the kind of mathematics to use. Mathematics does not always work. When we found that scalars did not work for forces, we invented a new mathematics, vectors. And going further we have invented tensors... Thus my second explanation is that we select the mathematics to fit the situation, and it is simply not true that the same mathematics works every place.”

Thus non-platonic geometrical patterns are deselected, according to Tegmark's platonic vision. That is, they don't fit the mathematics. Platonic shapes, patterns, etc., on the other hand, are “select[ed]” instead and other mathematics is used to “fit” more convenient “situation[s]”. In that case, those non-Platonic patterns or shapes aren't in Tegmark's “reality”. Doesn't that create a problem for the oneness of mathematics and reality? This also means that the “mathematics at hand does not always work”. At least it doesn't (if we follow the logic of Richard Hamming) until a new maths is utilised.

In addition, if particles, etc. are “mathematical objects”, then aren't we using mathematics to describe, plot or explain mathematics? Is mathematics, therefore, describing itself? Is it the case that we never get out of the circle of maths? Perhaps Tegmark likes that idea.

Platonic Structuralism

Tegmark's position seems to be a fusion of (ontic) structural realism and mathematical structuralism. Indeed, in one of his notes, he acknowledges John Worrall's structural realism thus:

“In the philosophy literature, John Worrall has coined the term structural realism as a compromise position between scientific realism and anti-realism; crudely speaking, stating that the fundamental nature of reality is correctly described only by the mathematical or structural content of scientific theories.”

As for platonic mathematical structuralism, Tegmark's most clear exposition of his position is the following passage:

“The notation used to denote the entities and the relations is irrelevant; the only properties of [for example] integers are those embodied by the relations between them. That is, we don't invent mathematical structures – we discover them, and invent only the notation for describing them.”

The platonic part of the above passage is expressed in the final sentence about our discovery of abstract entities. (That sentence doesn't logically follow from the proceeding words.) Tegmark also says that a structure is “a set of abstract entities” How can that be? Surely entities (or elements) are parts of structures. The structure may well be derivative of the relations of the entities (to each other). However, there's still a distinction to be made here. A set of abstract entities is, well, a set, not a structure. Unless Tegmark takes sets and structures to be one and the same thing.

Tegmark is also both a mathematical structuralist and a mathematical platonist. Indeed this position exists in mathematical structuralism itself and it's opposed to Aristotelian mathematical structuralism; though it's not necessarily identical to Tegmark's own position.

In terms of the specific platonist position: mathematical structures are deemed to both abstract and real. This position is classed as ante rem (“before the thing”) structuralism. The platonist position on structures can be characterised as the position that structures exist before they are instantiated in particular “systems”. The Aristotelian position on structures, on the other hand, has it that they don't exist until they are instantiated in systems.

To explain the platonist position one can use Stewart Shapiro's own analogy. In his view, mathematical structures are akin to offices. Different people can work in a particular office. When one office worker is sacked or leaves, the office continues to exist. A new person will/can take his role in the office. Thus offices are like mathematical structures in that different objects can take a role within a given structure. What matters is the structure – not the objects within that structure.

Nonetheless, the people who work in offices are real. The idea of an office which is divorced from the people who work in it is, of course, an abstraction. Thus one may wonder why the office/structure is deemed to be more ontologically important than the persons/objects which exist in that office/structure. Surely it could be the other way around.

Tegmark also seems to fuse his position with that of relational theory. Indeed the former (platonic) mathematical structuralism can be seen as being an example of relationism (or vice versa). For example, Tegmark says that

“[t]o a modern logician, a mathematical structure is precisely this: a set of abstract entities with relations between them. [He then cites the integers and “geometric objects” as examples.]”

And, on page 267, he sounds very much like Lee Smolin and Carlo Rovelli when he tells us [as quoted at the beginning] that

“the Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that then properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks.”

Tegmark's Examples of the Use of Numbers

Tegmark keeps on supplying us with examples of what numbers do and what functions they serve in physics. Yet he does so without going into much (or even any) detail as to why the world itself is mathematical. We're told about numbers “representing” letters in computers; how pixels are “represented by numbers”; how the “strengths of the electron field and the quark field relate to the number of electrons and quarks at each time and space”; etc. In more detail, Tegmark says

“[that] there's a bunch of numbers at each point in spacetime is quite deep, and I think it's telling us something not merely about our description of reality, but about reality itself”.

Yet Tegmark appears to contradict himself. At one point he says that a field “is just [ ] something represented by numbers at each point in spacetime”. Here we have both the words “something represented” and “is just”. More clearly, we have a field which is “represented by numbers” and “just is” - the latter two words implying that all we have is numbers. Again, Tegmark says that the magnetic field is “represented” by “three numbers at each point in spacetime”. Yet he doesn't say that the magnetic field is a set of numbers (or even a “structure” which includes numbers).

Tegmark also cites the example of y = x² describing the parabola and x² + y² = 1 describing the circle. Yet aren't these descriptions of shapes, trajectories, etc., not of numbers or mathematics? Don't the numbers plot the shapes, trajectories, etc., not equal them? Yet that's precisely what Tegmark denies.

In addition, one would be hard pressed to interpret that idea that “space and time themselves are contained within large mathematical structures” without additional information.

Tegmark then indulges in what amounts to number mysticism. For example, he asks: “[W]hy are there 3 dimensions, rather than 4 or 2 or 42?” He also asks: Why are there “exactly 6 kinds of quarks in our Universe?”. But couldn't we just as easily ask: Why are there 101 dimensions, rather than 4 or 2 or 42? Similarly: Why are there 109 kinds of quarks rather than 6?

What do these questions so much as mean? What would acceptable answers (to Tegmark) look like? True, because there are 6 quarks, then only certain (physical) consequences are allowed to occur. The same is true of 3 dimensions. But is Tegmark making another point here?

In addition, is spin or charge really “just a number”? Tegmark connects charge and spin number to lepton number; though the latter seems to belong to a different category. That is, the number of leptons is a very different thing to the numbers we assign to spin and charge – even if the numbers are the same. What we have here (as stated) is number mysticism of the most crude kind. (This kind of number mysticism can be most clearly seen - in my view - when it comes to correlating the Fibonacci sequence with aspects of nature; especially in view of the later Kitty Ferguson quote. See this titillating documentary here.)

Another point worth making is that if numbers can plot anything, that's simply because they can plot everything. That is, if mathematics can explain or describe random events, chaotic conditions, or dynamical systems (which it can), then it can also explain or describe just about everything.

What I mean by this is that it's always said that mathematics is perfect for describing or explaining the symmetrical, ordered and even “beautiful” aspects of nature. Yet the science writer Kitty Ferguson throws a spanner in the works in the following:

“The diamond shapes in a sunflower seed-head [are] lop-sided. One had to give tree-trunks the benefit of the doubt in most cases to call them cylinders. The earth bulges and is not a perfect sphere. Natural crystals are not perfect geometric shapes either.” As for mirror symmetry, one side of the human face is not the true mirror image of the other... At the level of elementary particles, we discover a right- and left-handedness about the universe, slightly favoring the left. In the early universe there may have been an infinitesimal imbalance between the amount of matter and the amount of antimatter, an imbalance which has resulted in the universe of matter we see today.... If someone or something had taken the symmetries found in physics and 'corrected' them, we and our universe could not exist.”

Ferguson then carries on with her theme:

“The things we build and the art we we create exhibit much more geometry and symmetry than we can find in nature. Are we bettering nature, imposing rationality on a less rational universe?

Roger Penrose also makes the point that mathematical descriptions of the world are often (or always) more “beautiful” than the world itself.

And Richard Hamming chimes in with the following statement:

“Humans see what they look for... our intellectual apparatus is such that much of what we see comes from the glasses we put on.”

In terms of my own take on this. If I were to randomly throw an entire pack of cards on the floor, then that mess-of-cards could still be given a mathematical description. The disordered parts of that mess would be just as amenable to mathematical description as its (accidental) symmetries.

Similarly, if I were to improvise “freely” on the piano, all the music I played could still be given a mathematical description. Both the chaos and the order would be amenable to a mathematical description and even a mathematical explanation. Indeed a black dot in the Sahara desert could be described mathematically; as can probabilistic events at the quantum level. It's even possible that mathematicians can find different – or contradictory – symmetries in the same phenomenon.

Another problem here is that many philosophers, mathematicians and physicists have said that mathematical physics doesn't use all of mathematics to describe the world. That is, there are large chunks of maths which seemingly don't apply to the world; or, at the least, at present they don't fulfill a purpose in mathematical physics. Yet if maths and the world are one, then why are there (to use Roger Penrose's words) “bodies of maths with no discernible relations to the physical world”? In terms of Penrose's actual examples:

“Cantor's theory of the infinite is one noteworthy example... extraordinary little of it seems to have relevance to the workings of the physical world as we know it... The same issue arises in relation to... Godel's famous incompleteness theorem. Also, there are the wide-ranging and deep ideas of category theory that have yet seen rather little connection with physics.”

Here again, according to Tegmark's position, this split can't be real. If mathematics and the world are one, then it doesn't make sense to say that there are parts of maths that aren't applicable to the world. Having said that, it still seems acceptable to argue that we can have parts of mathematics that aren't applicable to the world and yet still accept that the world is mathematical.

Conclusion

Is mathematics seen (if tacitly) as God's language? Did God write the “book” which Galileo referred to? Indeed is this assumption implicitly behind much of what Tegmark and others argue?

Of course the precise relation between mathematics and the world (or reality) has been debated for a long time. As we have seen, this isn't such a big problem for Tegmark for the simple reason that he believes that mathematics and the world are one and the same thing.

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