Chris Port Blog #264. A Classic Example Of How Empiricists Can Get Themselves Into Trouble... Which came first, maths or the universe?

© Chris Port, 26^th May 2011

The following discussion is a verbatim record of my involvement in a comment thread on The Richard Dawkins Foundation for Reason and Science (Official) Facebook page. It's a classic example of how empiricists can get themselves into trouble without metaphysical perspective...

S: "I have just had a thought that may bear some examination and may be a problem for theists, though they will have a facile way out. It has just occurred to me that Mathematics is something that exists in effect outside of the universe and simply is. Completely immutable, absolute, existent before time or the big bang, defining everything there after, and built upon rock solid axioms. Like “god” Mathematics is eternal and self existent though sadly has offered no opinion as to how Noah should have built the ark."

...

Me: "The word ‘exist’ may be leading you astray here. Mathematics is the study of quantity, structure, space, and change. It is overwhelmingly probable that these phenomena exist independently of our minds. However, mathematics is the study of the phenomena, not the phenomena themselves. Mathematics is the attribution of consistency in the mind. The universe itself is as indifferent to mathematics as it is to beauty. Without consciousness, mathematics would not exist."

S: "It would but without consciousness it would not be acknowledged, gravity at sea level was still around F=MA has always been the case, and the universe is mathematics made function. I see where you are coming from but these relationships or more simply say prime numbers, are there whether they impinge on our consciousness or not. And these relationship or rules will be consistent when and where ever, though F may not equal M A. And looking why back I pulled back from claiming that these relationships can exist pre existence, but would insist on them being latent. Any way it was only a thought I had while driving a bus around in circles earlier today. A possible atheist god if you like, so I'm not living or diieing on its verifiability."

"dieing" 

Me: "I think that you may be falling into a classic teleological trap (getting things back to front).

Teleology is the supposition that there is purpose or direction in natural processes. To say that “the universe is mathematics made function” is to infer that mathematics preceded the universe, and that the ‘purpose’ of the universe is to give mathematics physical form.

I would argue that in reality it is completely the other way around. The universe came first. Mathematics is our expression of patterns that we observe in it. These patterns are relationships attributed by our minds, not inherent properties. The only inherent properties of the universe are the four basic force charges and the ways in which they interact with each other. Everything else in the universe is an ‘emergent property’ rather than an expression of function."

‎"... 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." ~ Albert Einstein.

‎

S: "OK I give up, it was only a thought and I wouldn't dream of giving the universe function. But I won't budge on 1+1=2, when ever and where ever."

Me: "I once thought that I would never budge on Descartes’ “Cogito ergo sum” (“I think therefore I am”). Unfortunately, science has revealed this to be almost completely wrong! It would be now more correct to say “A thought, therefore something having it”…

I won’t argue with 1 +1 = 2. That’s true in as far as it goes. It just doesn’t go very far. It doesn’t mean anything (other than being a trivial axiom).

Logical consistency, as exemplified by mathematics, has some shaky claims on ‘truth’. 1 + 1 = 2 may be true, but that doesn't mean anything other than 1 + 1 = 2. To ask what it means is meaningless. You either accept it or you don't. If you do, it doesn't mean anything. And if you don't, then (mathematically) you can't say anything else. Kurt Gödel's incompleteness theorems prove (in the mathematical sense of the word) that there is no complete and consistent set of axioms for all of mathematics

Only in aesthetics and ethics will you find meaning - and (I suspect) it is meaning that people want from their truths, rather than mere pedantic consistency.

Maths is not a meaningful language because its notation has no subtext or context. It communicates nothing. However, its self-referential expansion to develop consistent patterns of logic (which some find elegant and beautiful, expressing infinite complexities in pithy formulas - like poetry but with numbers) hints at something deeper...

Truth is intangible. Like the humble electron, it is affected by our clumsy measurements. Once you realize that it cannot be grasped, only then have you truly understood it. Truth, like humanity, is inherently paradoxical, a probability waveform. Confused? If so, good. Like quantum theory, if you think you understand it, then you don't! That's a good starting place for science and art...

Truth, logic and consistency are human constructs. We attribute importance to them for various practical, moral and aesthetic reasons. Pattern recognition and consistency have evolved in our species because they were useful survival traits....

So, once, was religion. Whether it is now a help, a hindrance or a harm is currently the subject of much heated debate. The real question is whether science and religion are actually talking about the same thing.

Richard Dawkins is happy to allow artistic fiction to tell us meaningful truths about life (he is a cultured, aesthetic writer) but he is not happy to allow religion to do the same because (a) he believes it is a fiction posing as a truth, but more importantly (b) because he believes it is harmful. Art, if you like, is a ‘good’ lie and religion is a ‘bad’ lie.

Even in its benign, compassionate form, he believes that religious belief acts as the thin end of the wedge for irrational indoctrination (often without choice) and, ultimately, an intolerance of scientific methodology.

In essence, there is no middle ground or fence to sit on. I agree with most of what he says, but I do see some gaps in communication. We have a good old dialectic here: science versus religion, each making moral claims about what is ‘good’ for humanity.

Is there a synthesis, or do we have to jump one way or the other? I would plead for some breathing space to allow Wittgenstein’s ‘language games’ to develop a more ‘artistic’ dialogue. Stop worrying about ‘reality’ (whatever that may be) and start analyzing how we communicate."

T: "I'm short on time and my posts haven't been working here, but Chris, how do you solve the problem of unity out of the diversities when it comes to logic/math? If we are contriving these realities [individually] how do they work as a unity?

Me: "It’s a fascinating area of study. Does the recurrence of similar patterns in different systems suggest a metaphysical role for mathematics? The problem then would be an inference that physical forms are an expression of mathematical functions…

My initial solution to this ‘problem’ is simply to suggest that there is no problem!

Starting from the four fundamental forces, why shouldn’t localized systems arrive at similar ‘solutions’, even at different scales? The diversity of emergent properties is the result of slight inhomogeneities in the quantum state of the early universe. Time, expansion and chaotic (complex) interactions have provided a wide variety of phenomena for us to study. We use mathematics to model their forms and behaviour. Some physicists have even speculated that chaos theory may give rise to new ‘emergent’ laws of physics, but the gist of String/’M’ Theory seems to be that all matter and energy is just variations on a fundamental theme.

Unity in diversity only seems to be a problem if you think that this is an underlying ‘design’ feature. I would cite the Weak Anthropic principle and say “That’s just the way it is, otherwise we wouldn’t be here to see it”.

As a very simple example, many different types of forces tend to be transmitted equally in all spatial directions. One consequence of this ubiquitous behaviour is that the pattern of a circle or sphere is likely to recur throughout nature at different scales. If mathematics starts to develop self-consistent variations on the component relationships of circles and spheres, it shouldn’t really surprise us if we start to discover more and more apparent similarities in different phenomena. Basically, the more complex mathematics becomes, the more opportunities it creates to find patterns. That’s what maths is designed to do.

Whether this recurrence of patterns suggests anything deeper actually leads us away from numbers towards language and metaphysical discussions (which I suspect you wish to avoid!). For example, if I was to claim that the ubiquitous recurrence of ‘God’ in different localized cultures inferred that there must be something like ‘God’ in nature, I suspect you would retort that all it actually reveals is people’s general urge to invent explanations. A diversity of religions does not make a unity in a deity.

Once the concept of ‘God’ is invented, this can lead us into all sorts of teleological traps. I would say that ‘unity in diversity’ is actually just another variation on this theme. We invent a notation designed to find patterns. We then map that notation on to the physical universe. We are then surprised to find recurring patterns in the notation. I don’t think that’s a problem. I think that’s what we were looking for, so we found it."

See:

Mathematics - Unity in Diversity
http://pythonism.wordpress.com/2010/04/14/math-unity-in-diversity/

The Significance of Unity and Diversity for the Disciplines of Mathematics and Physics
http://www.metanexus.net/magazine/tabid/68/id/10892/Default.aspx 

Functions and Mathematics
http://www.biblicalchristianworldview.net/Mathematical-Circles/functionsMath.pdf

‎T: “Nice Google. Now, please explain how, if we are [individually] [making] mathematical truths, subjectively, how they are united into a coherent [objective] system."

Me: "Because that's what we've designed the system to do. If it's not consistent, its not maths. You're starting to slip into Wittgenstein's 'category errors'.

Mathematics (unlike language) is specifically designed to be consistent and coherent. The universe, by existing, is coherent. The ‘laws’ of physics are consistent and (quantum theory of gravity aside) mostly coherent down to the Planck scale. To map a coherent system onto a coherent universe, then claim that the universe is somehow a manifestation of the system, is to confuse two different categories of coherence. They are physically unrelated. They just look similar. Fortunately, the consistency and coherence of our mathematical system enables us to make testable predictions about the consistency and coherence of physical systems. There is a ‘family resemblance’ concept at work here. But they are not the same thing at all."

T: “You didn't answer the question, my friend;)"

Me: "I have answered it. I've just not answered it on the terms you've suggested ;)

Coherence gets more ‘fuzzy’ at the quantum scale. Quantum mechanics is the most consistently accurate scientific theory ever devised. Heisenberg was dismissive of attempts to understand what was ‘physically’ going on. As far as he was concerned, all that could be claimed about quantum mechanics was that the maths worked. The uncertainty principle, and the strange interference of measurement and even consciousness on quantum level 'events'/probability waveforms, are still profoundly incomprehensible to us. As a lyricist physicist, I derive a wry satisfaction from this. Maths is designed to be ‘perfect’, yet the universe (so far) eludes perfect notation. There is, of course, no such thing as an objective system.

T: “Great! Can't answer, so you pull the old QM card....gotta love it!"

Me: "All roads lead to foam...

Although (strictly speaking) non-sequiturs, you may find the following 'family resemblance' posts amusing when pondering some of the discrepancies between numbers and 'reality'...

Marty Gull - Targets
http://martygull.blogspot.com/2011/03/chris-port-blog-130-marty-gull-targets.html

Monkey Dust - Government School Targets
http://www.youtube.com/watch?v=zLvDKI1T14Q&feature=related 
 

My training is in aesthetics rather than mathematics. However, borrowing from perturbation theory, perhaps another way of clarifying the apparent subjectivity/objectivity ‘problem’ of unity in diversity is to transpose an aesthetic methodology onto an empirical impasse. I’m not suggesting that the two are directly analogous, simply that clarity of thinking in one discipline can sometimes lead to clarity of thinking in another discipline.

I'll post the different perspectives one at a time to avoid overwhelming you with data. We'll need to consider: Simple Objectivism, Simple Subjectivism, Simple Relativism, Sophisticated Objectivism, Sophisticated Subjectivism, and Sophisticated Pragmatic Relativism...

Simple Objectivism (Recognizing the rules of composition).

Simple objectivism depends simply upon the observer's recognition of the ‘rules’ of composition (consistency in this case) and not on any supposed feelings aroused by them. Indeed, the satisfaction or pleasure experienced by the observer is supposed to arise from their skill in recognition.

'Whatever the peculiar causal conditions… the … features…are themselves independently perceivable. This gives the ... [phenomenon] a certain critical autonomy.' (David Cooper).

However, Cooper then goes on to describe a basic problem with simple objectivism. The evaluative force of a judgment claims more than that the phenomenon possesses certain qualities. It also claims that the phenomenon merits attention and thus ascribes a value which is not inherent in the phenomenon itself. Where does this value come from? One possible answer is that, while the rules of the phenomenon may be objectively observed, the values given to it may arise from a simple subjectivism.

Simple Subjectivism (I know what I like but do I need to know why?).

According to simple subjectivism, the judgment does not depend upon the rules of composition in the phenomenon but in the pleasure or displeasure that perception of the phenomenon happens to arouse in the spectator. This implies that one spectator experiencing pleasure and another spectator experiencing displeasure from the same phenomenon would not be contradicting each other.

However, as Cooper notes: '... we are normally expected to try to show that the judgment rests upon features which render our response a justifiable one'. By what rules, to what indisputable court may we appeal to justify our judgment? Are all judgments only relative to each other?

Simple Relativism (Is one judgment as good or as bad as another?).

Simple relativism (often referred to as dogmatic or vulgar relativism) proceeds from the rather pessimistic premise that, since we cannot incontrovertibly prove universal judgments, all judgments must logically be equally valid (or worthless).

Cooper argues that this 'anything goes' approach is doubly inconsistent. It self-defeatingly allows for universalist theories and also implies the existence of a supposed universal standard of which all theories must fall short.

It is impossible to find a system which could possibly exist in a self-enclosed way; certainly it is theoretically difficult to define what it would look like.

It seems then that, in their simple versions, theories of objectivism, subjectivism and relativism are all flawed by their lack of predictability. Objectivism cannot predict values, subjectivism cannot predict judgments and relativism cannot predict anything. It thus remains for us to consider whether a finer, more sophisticated version of one of these theories might be threaded through the eye of our ontological needle.

Sophisticated Objectivism (All in the imagination?).

Sophisticated objectivism was developed by the eighteenth-century philosopher Immanuel Kant in his Critique of Judgment. In some ways, it is a fusion of simple objectivism with simple subjectivism.

While Kant claimed that objectivist judgments demand agreement from everyone without exception, he also allowed that the determining factor of some viewpoints was the feeling of contemplative pleasure or displeasure aroused in the mind of the spectator (a simple subjectivist viewpoint).

Kant linked these two seemingly mutually exclusive viewpoints by giving this paradox the structure of an 'antinomy' (a contradiction existing between two apparently indubitable propositions).

The key to understanding how these opposing viewpoints can co-exist is in Kant's emphasis on the 'free play' of the imagination. In essence, this means that the spectator's imagination is free to introduce concepts which are not inherent in the phenomenon and then rationally defend such a judgment by reference to these concepts.

An important consequence of Kant's 'free play' of the imagination is that different spectators can only be perceiving the same phenomenon in so far as they possess the same faculties of understanding and their imaginations operate identically.

However, it is difficult to refer to any particular attribution to justify one's assessment other than a sophisticated subjectivist perspective of what is pleasurable or displeasurable.

Sophisticated Subjectivism (Colours and aesthetics - 'seeing red'?).

Sophisticated subjectivism was skilfully defended by the eighteenth-century philosopher David Hume. In Of the Standard of Taste, Hume drew an analogy between colour judgments and aesthetic judgments whereby even the subjectivist occurrence of colour in the observer's mind still allows for standards in assessing the appropriateness of particular colour judgments and the capacity of the observer to make such a judgment.

For example, we would be sceptical of the colour judgment of someone who was known to be either blind or colour-blind.

It is worth remarking here that it was correct for Hume to attribute the occurrence of colour to subjectivism rather than objectivism. As the twentieth-century philosopher Bertrand Russell showed in The Problems of Philosophy, we cannot say that objects 'possess' colour. Colour is simply the result of light-waves in one part of the spectrum being reflected off an object and into our eyes while light-waves in another part of the spectrum are absorbed either by the object itself or by the intervening medium. In astronomy, physicists use this technique (known as spectroscopy) to deduce the characteristics of interstellar phenomena.

While there is a relatively uncontentious agreement on colour judgments, this does not so clearly hold true for other judgments. Even if it did, this would not be equivalent to a claim of universality. By its very nature, a subjectivist mode of assessment must allow for the possibility that not all spectators will arrive at the same judgment.

'Unlike the objectivist, a defender of subjectivism, including the sophisticated variety, cannot maintain that a judgment, if correctly made, must hold for everyone without exception... To those of us whose sensibilities may happen to be governed by totally different principles from the majority's - always a possibility for a subjectivist - the judgments of discriminating spectators within that majority can have no logical force.' (David Cooper).

Pragmatic Relativism ('Local' truths).

The problem with simple or dogmatic relativism was that, in adopting a form of Nietzchean nihilism, it inadvertently implied that there ought to be a set of universal criteria which all theories fail to meet. How can we fail a theory for failing to include a non-existent quality?

By way of contrast, pragmatic relativism does not condemn any one judgment for failing to be universal. Nor does it imply that any judgments should be universal by giving a despairing, post-modernist equivalence to all judgments. It simply recognizes the de facto existence of different judgments.

While judgments are relative to distinct practices, they are not objectivist (they do not demand agreement from everyone) and they are not subjectivist (they can be defended as impersonal, localized 'truths').

The main attraction of pragmatic relativism is that it is an essentially modest theory (it does not assert its supremacy over other useful theories) and allows us to be receptive to competing judgments by emphasizing their differences rather than their similarities.

The main failing of pragmatic relativism is that, in failing to acknowledge the superiority of any one mode of judgment (including itself) it cannot arbitrate between competing judgments.

Therefore, while pragmatic relativism may be useful in discussing different perspectives, it is unable to provide us with any conclusion.

Despite the inherent inconclusiveness of pragmatic relativism, it is a nimble enough perspective to enable us to avoid the logical pitfalls which open up before each of the other perspectives discussed.

For myself, I would therefore proceed to examine the apparent phenomenon of unity in diversity using a sophisticated pragmatic relativistic perspective rather than an empirical one. Perhaps mathematics (and, annoyingly, religion), have an underlying aesthetic overlap? Truth is beauty and beauty is truth. Then it gets a bit complicated..."