The Mathematical Universe
On this page we will consider the rather uncanny central role that mathematics appears to play in the structure of the universe. We will also consider the theory that certain mathematical structures have a form of "reality" all of their own.
It could be said that the role mathematicians is to discover truths which are already "out there". These truths are no inventions of clever men - it does not matter who invented the mathematical structure of complex numbers, for example. Such structures have been there since the beginning of time - an eternal truth - waiting to be uncovered.
This view that certain mathematical concepts are eternal truths which apparently have an independent reality of their own was proposed in ancient times by the great Greek philosopher Plato (c. 360 BC). Consequently, the mathematical structures with this apparent reality are called Platonic.
And perhaps the most stunning example of a mathematical structure which has recently been discovered is the Mandelbrot Set.
The Mandelbrot Set
It was Benoit Mandelbrot who first introduced us to the beauty of the Mandelbrot set in 1980. However, when Mandelbrot first saw the strange patterns printed by his computer he suspected his computer was broken!
The Mandelbrot Set is produced by a remarkably simple mathematical formula - a few lines of code describing a recursive feedback loop - but can be used to produce beautiful, coloured, computer plots. What makes it so extraordinary is that it is possibly to endlessly zoom in to the set revealing ever more beautiful structures which never seem to repeat themselves. It's almost as if it is a mathematical object with an independent existence of its own, and we are the explorers investigating this uncharted mathematical world.
Referring back to our earlier discussion, it might be thought that the Mandelbrot set has a Platonic reality all of its own - it doesn't matter who creates the diagram, or which computer is used, the structure will always appear the same. To quote Roger Penrose from The Emperor's New Mind: "The Mandelbrot Set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot Set is just there!"
The applet below allows you to explore the Mandelbrot Set for yourself.
Mandelbrot Applet InstructionsYou can zoom into the image below by creating a
window around the area you want to zoom to. To do this, just click on the image
and drag a box (Tip: try zooming in to the very narrow region
between the red and the black areas).
Just click once to reset to the
original image.
(Applet
created by Andreea
Francu)
I tried playing around with the applet above and produced the images below. See if you can do better:
Using the applet I discovered the image on the left below. Don't you think it looks like a piece of coastline?
The reason for this is that the border of the Mandelbrot Set has a shape like a fractal (a fractal coastline actually has infinite length, revealing endless detail as you zoom in. For more information see the section "The Tower of Turtles" on the It's a Small World page). Fractals are geometric shapes found throughout nature which are characterised by having many similar branches. Fractals are, in fact, self-similar because no matter how far you zoom into them they still resemble the original object (a branch of a tree, for example, resembles the whole tree):
So here we can see an example of how mathematics is an underlying force in the design of nature. Simple mathematical rules are responsible for the beautiful complexity of nature. Maverick genius Stephen Wolfram stresses the importance of mathematics in designing nature in his controversial book A New Kind of Science (see this Forbes article). We have a tendency to assume that evolution is the sole factor in designing nature. However, Wolfram reveals the undoubted role of mathematics alone in designing not only trees but also objects such as sea shells:
The fractal "Pascal's Triangle" creates the pattern on sea shells
(picture from Stephen Wolfram's A New Kind of Science)
Just as a sidethought, I sometimes wonder if the complex structure of the Mandelbrot Set can provide valuable insights into the nature of our universe (after all, the universe appears to follow mathematical principles). Whenever we talk about the Mandelbrot Set we are used to seeing beautiful, colourful pictures of swirling complex patterns. But what is not generally realised is that the wonderful complexity can only be found in the very thin border region of the set. The iterative equations used to produce the set can be used to assign a colour to every pixel in the entire (infinite) 2-dimensional Argand plane (i.e., treating the Cartesian coordinates to represent complex numbers), but the black area which stretches outside the colourful area is completely blank (the "Big Boring Area" in the diagram below). Similarly, the area inside the Mandelbrot border is completely blank (the "Small Boring Area" in the diagram below). Only the very thin border between these two regions contains the fractal complexity which has made the Mandelbrot Set so famous (the "Interesting Bit" between the two red arrows below).
Maybe we could imagine the complex structure of the Mandelbrot Set as a "mini universe". Then it might be possible to find an analogy between these three regions of the Mandelbrot Set and our universe. Let's try equating distance from the centre of the Mandelbrot Set with increasing scale in our universe. The vast, unbounded region on the outside of the set would equate to the largest scales (e.g., galaxies), whereas the very smallest scales toward the centre of the set would equate to elementary particles. The thing is, it could be said that the most interesting thing in the universe (intelligent life) occurs in the very narrow fractal border region (human scale) between these two (rather predictable and boring) extremes of scale.
Copernicus discovered that the earth does not hold a special position in the universe - but maybe he was considering the wrong "space" (John D. Barrow from The Infinite Book: "While Copernicus's idea that our position in the universe should not be special in every sense is sound, it is not true that it cannot be special in any sense"). If we move away from considering our positioning in physical space and instead consider the positioning and conditions of Earth in an abstracted mathematical space then we would find it to be very special indeed. Consider a number of variables describing the conditions and positioning of Earth, such as the scale of the planet, its distance from the sun, its surface conditions, the positioning of the neighbouring planets, and then consider each of these variables in an abstracted mathematical space. For example, the diagram below plots "Mass of star relative to Sun" against "Radius of orbit relative to Earth's":
In that case, if we stood back and considered our findings for all our variables we would find that Earth's position and conditions are very special indeed: it's in a very narrow region called the Habitable Zone (indicated as a curvy grey band on the diagram above). Two red arrows are superimposed on the diagram to indicate that the Habitable Zone is analogous to the Interesting Bit shown on the Mandelbrot Set diagram considered above. In fact, you could say the Earth is right in that interesting fractal border region of the Mandelbrot Set where interesting things happen (see Wikipedia on planetary habitability hypothesis). The most interesting thing in the universe (intelligent life) occurs in this very narrow fractal border region, just as if it was a galactic Mandelbrot Set.
If I was an observer of the universe, that's the region I'd zoom into.
Mathematical Platonism
Let's return to the discussion on mathematical Platonism from the top of this page, the idea that mathematical concepts are eternal truths which have an independent reality existing outside of space and time. What sort of reality are we talking about for these mathematical concepts? Well, we're not talking about a physical reality which we can touch (see Tegmark on "radical Platonism" here - page 14). According to Roger Penrose in The Emperor's New Mind (which firmly advocates some form of mathematical Platonism): "It is the Mandelbrot Set's 'mathematician-independence' that gives it it's Platonic existence". So it's definitely different from the more usual concept of "hard" reality. Penrose goes on to declare: "My sympathies lie strongly with the Platonic view that mathematical truth is absolute, eternal, and external, and not based on man-made criteria; and that mathematical objects have a timeless existence of their own."
The remainder of this page will examine this claim that mathematical structures have some form of "reality". We start by considering the foundations of mathematics.
Mathematical proofs are built-up from simple axioms: self-evident truths. These simple axioms are combined via the laws of logic to create more complex theorems. These simple axioms can appear so obviously correct that the Wikipedia article on logical axioms goes as far as saying: "these are statements that are true in any possible universe" (see here). So maybe we can define axioms and the laws of logic - and the laws of mathematics which are derived from them - as that which could not possibly be any other way. If so, then we could indeed consider these particular axioms and laws as holding some preferred position in a Platonic realm.
However ...
Multi-Valued Logic
Modern mathematics can be treated as a formal system, formalised to such as extent that mathematical objects can be replaced by abstract symbols without particular meaning, at which point the subject of mathematics becomes completely divorced from physical reality. Formalist mathematicians are then free to select an arbitrary set of axioms and laws of logic (not limited to the "obviously correct" axioms of the Platonists) as long as the resultant systems are consistent.
(It is essential that any mathematical system built-up from axioms is consistent, i.e., no statement in the system can be both true and false, the system contains no contradictions. If that was the case then the whole system would collapse as it would be possible to prove any statement to be true. If you've got an inconsistency then your system is basically broken. Apparently contradictory paradoxes such as Russell's paradox (considered later) also introduce inconsistencies into the system.)
The formal systems approach when applied to the field of logic reveals that traditional two-valued (True/False) logic is not the only possible form of logic. For example, let's consider Aristotle's Law of the Excluded Middle (see here) - an "apparently obvious" law of logic which states that "either a statement is true, or the statement is false; no other condition is possible". At first glance it might appear obvious that a proposition can be either "true" or "false", with no other value being possible. But it is possible to imagine a third state of "unproven" or "uncertain". For example, it is not yet possible to determine if the statement "It's going to be a white Christmas" is true or false, at this moment in time we have to regard the statement as "undecided".
Don't dwell to long on this specific "White Christmas" example. The general point being made here is that it is possible to conceive of logical systems containing three states - whatever those three states might be. In fact, it is now known that there is an infinity of different consistent logics with an infinite number of possible truth values.
This result raises doubts as to whether axioms are really the same "in any possible universe". The question might be asked: why was two-valued (bivalent) logic apparently selected as the dominant force in our universe? Why do we not inhabit a universe in which three-valued (ternary) logic dominates? This notion of specific mathematical and logical structures apparently being "selected" to play a dominant role in our universe apparently gives weight to the Platonists' argument of there being a select realm populated by "perfect" mathematical objects, with certain axiomatic and logical systems apparently occupying exalted positions.
However, on closer analysis, we see that although two-valued logic dominates, there is still a role for three-valued logic (such as in the "White Christmas" example). No doubt we could imagine situations in which four-or-more valued logic would be appropriate. So maybe two-valued logic does not hold such an exalted position. Two-valued logic dominates because the vast majority of propositions can be determined with certainty (i.e., true or false), but this does not mean there cannot be roles for other mathematical structures, axioms, and logic in our physical reality. They're just not so common.
As John D. Barrow explains in his comprehensive study of mathematical Platonism, Pi in the Sky: "The attempt to create a heavenly realm of universal blueprints that are truly different from the particulars founders under the weight of another simple consideration. Plato wants to relate the universal abstract blueprint of a perfect circle to the approximate circles that we see in the world. But why should we regard 'approximate' circles, or 'almost parallel lines', or 'nearly triangles' as imperfect examples of perfect blueprints. Why not regard them as perfect exhibits of universals of 'approximate circles', 'almost parallel lines' and 'nearly triangles'? When viewed in this light the distinction between universals and particulars seems to be eroded". No mathematical structure can claim superiority over any other structure.
To sum up, the Platonists' claim - that certain mathematical structures which play dominant roles in our physical reality have an exalted position in some Platonic realm - does not really stand up to investigation. It's easy to get fooled by beautiful, complex mathematical structures. And when "real" objects (such as humans and computers, or any physical process) come along they can endow an illusory, deceptive reality to those structures (such as the Mandelbrot Set). It's easy to get taken in and entranced by this "grand illusion" (read Stanislas Dehaene on the cognitive illusion of mathematical Platonism here).
Why does Physics follow Mathematics?
The principle of axioms might go some way to explain the almost uncanny match between mathematics and physics. Mathematics has been almost uncannily useful in explaining the natural sciences. It is almost weird the way that developments in mathematics, and the discovery of new mathematical structures, has been later matched by discoveries in physics which involve the similar structures in the physical world (see The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner, and the entertaining short story Unreasonable Effectiveness by Alex Kasman). For example, in 1931 Paul Dirac predicted the existence of the positron purely by considering mathematics - the first time a particle had been predicted from purely mathematical considerations. For this reason, mathematics has been called the "language of nature".
In order to get a better understanding of this apparently uncanny match, we should consider the origin of mathematics. Right back when man first started counting physical objects and discovered the usefulness of numbers and arithmetic for commerce, mathematics has been developed as a useful tool when it is used as an abstraction from physical reality. Maybe we now find a situation whereby developments in mathematics are later being shown to have equivalent counterparts in physics, but we would do well to remember that at the start it was physical reality that provided the motive for developments in mathematics.
It should also be realised that physical structures and processes are axiomatic systems - at their most fundamental level they are themselves also subject to physical axioms, self-evident truths. The axioms can be combined to describe the behaviour of larger, more complex physical systems: macroscopic behaviour results from microscopic behaviour (read about the building of a ladder of effective theories on the It's a Small World page). Indeed, David Hilbert wanted to see a full mathematical treatment of the axioms of physics (see here). In particular, he considered the kinetic theory of gases in which the pressure, temperature, or volume of gases could be found by considering the statistical mathematical behaviour of smaller constituent molecules. Paul Benioff has described a framework for linking physics and mathematics: "The basic properties of the physical and mathematical universes and a coherent theory are emergent together and mutually determined and entwined" (arXiv paper quant-ph/0201093).
If mathematics was developed as a model of the behaviour of physical reality, and if physical reality is an axiomatic system, then it should be no surprise that the resultant mathematics turned out to be an axiomatic system. From the very start, mathematics was designed to match physics, to be an effective tool. Developments in physics provided the motive and inspiration for developments in mathematics. As developments in physics are now stalling (with the requirement for ever-larger particle accelerators) it's little wonder that developments in mathematics have forged ahead and are later found to mirror developments in physics. The use of symmetry in discovering new elementary particles has been especially remarkable, but it should be remembered that the initial inspiration for exploring mathematical symmetry came from exploring the natural beauty of the macroscopic physical world.
So we should not forget that the strength of mathematics lies in its ability as an abstracted tool for describing physical reality, which was its initial function. Hence, mathematics emerges as an epiphenomenon - see here. It could be said that mathematics has moved away from this role to consider structures which have reduced roles in physical reality, such as Gödel's Theorem and Cantor's "uncountable infinities". As explained in the previous "White Christmas" discussion, it might be possible to find uses for these results in physical reality, but that would prove a difficult task. Indeed, the mathematical principle of "infinity" appears to have no manifestation in physical reality, and the presence of infinity in physics theories is generally taken to represent a flaw in the theory (the elimination of troublesome infinities is a major reason behind the popularity of string theory). And the discovery of some logical paradoxes involving circular reasoning would also appear unable to correspond to any situation in the real world - see Russell's paradox (physical reality contains no such circular paradoxes: only if time travel was ever discovered might we see the emergence of "killing your own grandfather"-type circular paradoxes). As mathematics continues down this esoteric path, it might be considered to be "disappearing up its own backside": mathematics for mathematics' sake.
Mathematics in the Multiverse
Andrei Linde has considered the development of mathematics in different universes of a "multiverse" structure (Section 11 of arXiv paper hep-th/0211048).
Each "bubble" universe in an eternally inflating multiverse can have different values for the physical constants, and even different numbers of spatial dimensions (see the page on The Anthropic Principle for a discussion of eternal inflation and the multiverse).
In the discussion of Andrei Linde there is once again an implicit acknowledgement of the importance of mathematics as being first and foremost an abstraction from physical reality (no Platonic realms in Linde's world). Hence, the different physical conditions in each universe would result in different mathematical systems. Many universes (maybe possessing more or less than 3 spatial dimensions) would have physical conditions completely averse to the formation of stable structures (such as galaxies and planets) and the evolution of life would be impossible. Many universes would be unable to expand, or would rapidly shrink back to nothing. Mathematics, being an abstraction from physical reality, would be of no use in such a degenerate universe. Only in a stable, 4-dimensional universe such as our universe would mathematics be useful for analysing physical reality. Hence, the reason why we find mathematics such as useful tool.
Comments
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I have to say I don't agree
that either Platonism or formalism are somehow opposed to physical
interpretation - or at the very least I don't understand what philosophy you
propose would make these connections to physics clearer. If it is true that
physics somehow trails mathematics, then we obviously require some other
standard than physicality to judge what mathematics is "good". And I think you
are underselling the relevance of even very esoteric mathematics to physical
reality. Note that it might be better to say that quantum mechanics is based not
on a 2-valued logical system, but on one with infinitely many possible states,
since it predicts only probabilities. There is also the interesting sum of
positive integers:
1+2+3+4+...
Clearly this series diverges, but
it turns out that there are several ways of defining sums (based here on the
Riemann zeta function - pretty darn "abstract" mathematics) which give the
result:
1+2+3+4+...=-1/12
which actually turns out to have
physical significance in certain experiments.
Perhaps by defining your
philosophy more clearly you can sharpen your criticism of other mathematical
philosophies.
Kevin Saff http://kevin.saff.net/ - Kevin Saff, 9th January 2007
Hi Kevin, thanks for your
interesting comment.
My understanding of formalism is that it is
entirely distinct from physical interpretation - it is completely divorced from
the physical world. It's just symbols and their relationships. The choice of
initial axioms is entirely up to the discretion of the mathematician, and so
need not match the "obviously evident" (according to the Platonists) axioms of
physical reality.
My views on formalism are shaped by "Pi in the Sky".
Here's some quotes: "In formalism it should be possible to manipulate the
mathematical symbols according to the rules without associating an physical
'meaning' to the symbols. When the meaning of the symbols has been distilled
off, it is a formal system, free of interpretations in terms of concrete
objects." And a pretty damning criticism on page 141: "Is it an adequate
response to the mystery of the harmony between the mind, the physical world and
the symbolic relationships of mathematics? I think that we must find it wanting
in many respects. The arid nature of formalism - its divorce from the
application of mathematics to the real world and its removal of meaning from
mathematics - were high prices to pay. But whilst they might have been palatable
in return for the original goals of completeness and consistency, one seems to
have ended up with the worst of all worlds: a view of mathematics that sees it
as a lifeless game that we must simply hope is consistent" (referring to Godel).
You make a good point about quantum mechanics in that the Hilbert space
can have a basis composed of infinitely many vectors. However - and this is key
- we do not see those infinitely many possible states in physical reality. Once
an observation is made we only see one eigenstate in physical reality - after
the "quantum jump". So we don't encounter the infinity in physical reality. See
http://www.ipod.org.uk/reality/reality_wavefunction.asp
Similarly with your other good point about the Riemann zeta function.
Yes, this is an infinite series but, as you point out, it converges to a finite
value. And it is this converging property which makes it useful for
regularization in quantum field theory (see http://www.ipod.org.uk/reality/reality_small_world.asp
), producing a finite value which has applicability in physical reality. Yes,
infinity can exist in mathematics, but I don't see it in physical reality. You
can't have objects which are infinitely large, you can't travel at infinite
speed, etc. (actually, according to Paul Benioff, even "really large numbers,
such as those of the order of 2n where n = 1020 with
correspondingly large complexities, seem to play no role in arithmetic or
physics" - see http://arxiv.org/abs/quant-ph/0201093
). Maybe the universe is infinite in extent, and who knows what goes on in black
holes, but we don't seem to have infinity in physical reality. Regularization
eliminates the infinities so that mathematics can become real. - Andrew Thomas, 10th January 2007
After reading this article and
comparing the idea with one of my own theories, I think I found a unique
relationship which is profound.
I think of the universe as being
self-similar. I was trying to figure out the relationship between different
scales (bubbles or dimensions of space) when I noticed a pattern. If the size of
dimensions are exponential in nature (as appears to be the case in http://www.amherst.edu/~rloldershaw/NOF.HTM
via a scaling factor) and you map the center and edge of a bubble to 0 and 1 on
a logarithmic scale, then life seems to occur at the coordinates 0.5, 0.5, 0.5,
0.5... etc. In other words, when you map the distances, sizes, and complexities
of objects from 0 to 1, we are at the 0.5 zone, what you call the "interesting
bit". The 0.5 zone for atoms is Iron, which rests on the middle group of the
middle period of the periodic table. Fission and Fusion push atoms toward this
element, confirming its importance. Our galaxy, when structurally compared to an
atom, has S (the core), P (the bars), and D (the spiral arms) orbital
equivalents; it is the 0.5 of galaxies. The sun is of midrange size and distance
from the center of the galaxy, a 0.5 star. Our planet is perhaps also of
midrange size and distance from the sun, the 0.5 of planets, and is composed
mainly of Iron, the 0.5 of atoms. Coincidence? I think not.
You can read
some more of my ideas and related things on my page at http://www.bmfusion.com/
Chris -
Chris Winn, 9th April 2007
In my view, mathematics is an activity we engage in in order to understand and cope with our world. The structures and processes of the physical world are not the same as their mathematical formulation; the map is not the territory. Other activities such as music, art, technology etc also contain valid descriptions of the world. In the Rosen diagram above, the mathematical system is depicted as being a sort of one-to-one image of the natural system. However, one could exchange the Theorems and Inferences on the Mathematical side of the model for any human activity, say, Notes and Scores for a Musical encoding and decoding of the universe. Of course each modelling system has its own terms, functions and relations. What sets mathematics apart, perhaps, is its predictive power, which is why it so revered as a model for reality. But once again, any model, however powerful, can only reflect an aspect of the universe, not the full picture, which, in all likelihood, can never be known in its entirety. - Jeremy Becker, 20th April 2007
Very nicely put. Thanks. - Andrew Thomas, 20th April 2007
I think Gödel's Theorem has in
fact a profound link with physical reality: the lack of an absolute system of
reference as proven by Relativity leads ultimately to the same paradoxes
involving circular reasoning.
What about The Numbers being the ultimate (and
only) reality, like Pythagoras said? - Marian,
29th September 2007
An interesting point, Marian. Maybe there is no distinction between physical reality and mathematical reality (as Max Tegmark has suggested http://arxiv.org/abs/0704.0646v1 and Anton Zeilinger http://www.ipod.org.uk/reality/reality_zeilinger.pdf ). I'm planning to rewrite this page at some point (it's pencilled-in my schedule a year from now!) to include more about this idea that "there is only information". I still think Godel's Theorem is the most over-hyped piece of obscure mathematics ever - it's sole function seems to be to produce endless popular science books about it! On a more prosaic point, how on earth did you manage to enter the umlaut in "Godel"? I can't do that! - Andrew Thomas, 29th September 2007
Hi Andrew,
With your
permission, let's try to detail some difficult stuff:
1. What is the status
of Maths?
I agree with your point that "Gödel's Theorem is the most
over-hyped piece of obscure mathematics ever", but just that. Let Gregory
Chaitin spoke: “Gödel's incompleteness theorem tells us that within mathematics
there are statements that are unknowable, or undecidable. Omega –Chaitin's own
discovery, an irreducible, infinitely complex number- tells us that there are in
fact infinitely many such statements… something we cannot deduce from any
mathematical theory.” And more: “To put it bluntly, if the incompleteness
phenomenon discovered by Gödel in 1931 is really serious — and I believe that
Turing's work and my own work suggest that incompleteness is much more serious
than people think — then perhaps mathematics should be pursued somewhat more in
the spirit of experimental science…” (Omega and why maths has no TOEs; http://plus.maths.org/issue37/features/omega/
).
What we must hold is that Mathematics cannot be complete, cannot be
closed in a final theory, that all theories, not only strictly mathematical
ones, must have a limited power of prediction given by the quantity of
information contained in their sets of axioms and rules of inference (Chaitin,
Gödel's Theorem and information; http://www.cs.auckland.ac.nz/CDMTCS/chaitin/georgia.html
), that Maths must be pursued more as an experimental science, like Physics, and
that mathematical objects must not be seen as similar to Platonic 'noeton',
ideal things, but to mundane, physical things. And from the incompleteness of
Mathematics goes that of the knowledge, most generally speaking, because Maths
are the 'sine qua non' condition of all systematic knowledge.
2. What is the
status of Physics?
The status of Theoretical Physics cannot be, by no means,
better than the status of Maths; one can attain only a precision lesser or equal
with that of the tool he uses. The consistent theories of Physics cannot be
complete, they have limited power of prediction. (It will be very odd physical
theories be logically better than mathematical ones!)
For example, from
Descartes to Einstein, one basic issue in physics is the reference frame
(system), which enables the mathematical approach to physical problems. The
Theory of Relativity stipulates no absolute frame of reference (the absolute
space, 'ether', and absolute time from the previous theories are proof by
Einstein to be pure idealizations, they simply don't exist). Being related only
one to each other, not to an absolute referential, the logic of the set of all
the physical frames of reference lacks consistence and ends in a kind of
circular reasoning. The frames of reference relate/'support' one another, but,
the whole edifice is, metaphorical speaking, 'in air'.
(to be continued) -
Marian Radulescu, arches_saccas@yahoo.com, 4th
October 2007
(continued)
What about the
physical reality itself? The physical things, obvious, don't have absolute
existence. As expressed in Nagarjuna's philosophy (and one must not necessary be
a Buddhist to agree with this statements of good sense): “all phenomena are
without any 'svabhava', literally ‘own-nature' or ‘self-nature', and thus
without any underlying essence; they are 'empty' of being independent” http://en.wikipedia.org/wiki/Nagarjuna
).
3. What is the relation between Maths and Physics?
We saw there is
strong analogy between the two. It is like both, the things of nature and the
concepts of knowledge, suffer by a syndrome of imperfection (incompleteness).
The common (and in my opinion false) idea is that math, along with the whole
knowledge, ‘reflects' the physical world. (Almost all common idea prove, sooner
or later, to be false!)
What about an reverse, Pythagorean, relation: the
mathematical objects are the invisible building ‘material' for the manifest
natures of the physical world and consciousness? All is information… (maybe,
we'll talk about that with another occasion).
4. Why is that?
The
primary cause for this state of logical and ontological illness (circular
reasoning, infinities, incompleteness, imperfection, etc.) is the absence of the
absolute natures, or, at least, of an absolute reference in the manifested
world. Nor the physical existence, nor the knowledge can't be completed, closed,
‘perfect', in the absence of the absolute manifested as absolute. There is not
such a ‘thing'.
But, we wonder, could physical nature and consciousness
exist, as we experience them, just like that ‘in air', without the absolute
prop, without an ‘urprinzip'?
I think, not. The existence of the Absolute is
necessary. But, being ‘absolute', must be unique, more, must be the only
existence (if there is another, then both are not absolute), and this seem to be
in total contradiction with our experience.
5. The solution?
The
Absolute is the unmanifested substrate, permanent, unchanging, complete ‘real
reality'. Substrate for all manifested reality, which is changing, limited,
incomplete, with an inferior degree of existence. Absolute is also ‘ineffable',
'aporreton' (Damascius, the last of the Neoplatonists), that is, nothing can be
known and/or expressed in words about (if other, that is not the really absolute
– "The Tao that can be told is not the permanent Tao; the name that can be named
is not the permanent name." — the Dao De Jing).
This is the Meta-Physics.
Maybe, with another occasion, about how this ideas might apply to the Quantum
Mechanics.
In the end, the easy stuff: the umlaut... I just copy-pasted
the whole word from your main text up-page. - Marian Radulescu, arches_saccas@yahoo.com, 4th October
2007
Wow. Thanks for that super contribution, Marian. - Andrew Thomas, 4th October 2007
Thank you for your initiative, Andrew. - Marian, 4th October 2007
You keep saying that Math
(thinking about and studying mathematical concepts) was "designed" to describe
reality. Math (and our ability to practice it) wasn't "designed" to do anything
(unless it was designed by God, in which case you still don't know how God came
to practice Math). It simply arose in the minds of intelligent human beings and
other species, no doubt shaped by evolutionary effects, or by the will of God,
whichever you prefer. Experiments have shown that newborn babies are born able
to count (the ability improves with age, but newborn babies can count), and also
that monkeys, birds, and all sorts of animals can count up to small numbers.
Saying we developed Math as an abstraction of the physical environment is
inaccurate; it makes it sound like we conciously developed the basics of Math (I
really mean the basics) to suit our needs, i.e. we could have chosen not to
develop any Math at all if we had not wanted to. But this is false; one has no
choice but to be born with the capacity to count.
In fact, for all we
know, Math ability (i.e. the ability to count and that which may follow from
that ability) arises simultaneously with or before conciousness.
Physical problems have motivated much of the developement of Math,
concious developement by humans in order to solve physical problems, but they
did not motivate the *start* of Math. They were motivation *near* the beginning,
but they were not there *at* the beginning. The ability to count is developed
before birth, and does not have a concious motive.
The all-singing
all-dancing Math we know today has been motivated largely though not entirely by
physical considerations. However the most basic Math, arithmetic, arises/arose
at birth with no human motivation whatsoever, and in principle all mathematics
(or certainly all physically applicable mathematics) could be developed from it.
Properly considered, the question "Why did Math develop?" does not have
the answer "because we decided so, in order to suit our needs". We develop our
basic Math capacity before we do or decide anything, and the deliberately poor
answer given above does not take account of this. The answer is about the
physical universe and how evolution has shaped us or how God has created us,
whichever you prefer. Our minds are intrinsically mathematical, and our minds
are part of this universe, making it mathematical to some degree.
You
could then say "We merely evolved Math ability due to our situation in the
physical world". But that then leaves the unanswered question of why the
physical world would force such a chain of events. So you still have an
unanswered question.
As for disappearing up it's own backside, let's
just say if it were not for much of the crap coming out of it, much of modern
science and most of modern physics (relativity, QM, etc.) would not have arisen.
And Godelian considerations helped give rise to computers and modern computer
science. - Dan, 3rd December
2007
As for Godel, his work is
simply not overhyped. It is misunderstood, definitely. It does not say there are
statements that we cannot prove and it does not say that math must become
doubtful (that would be a question of certainty and uncertainty, which are
emotions, no mathematical theorem will compel one to feel either one emotion or
the other).
It says that for any formal system extending minimal
arithmetic there is an arithmetical statement which can be expressed but not
proven by that formal system. That is not to say that there are not other axioms
we can recognise as true, add to the system, and obtain a proof of the
statement.
Godel's work was brilliant not just for his theorem, but how
he revealed that provability of this or that theorem based upon these or those
axioms was literally a problem of the theory of numbers. Similarly a statement
of the theory of numbers can be recognised as a statement of the kind I just
mentioned in mathematical logic. This is a profound insight and one that Godel
developed to a high degree.
As for Chaitin, he was not the first to show
that infinitely many undecidable statements exist for each formal system, that
followed from Godel's work. Chaitin constructed, for a theory T, an infinite
class of arithmetical statements which corresponded to each bit/digit of a
certain real number omega, such that each bit was a 0 or a 1, and for each bit
it could not be proved in T what that bit was. He then showed that if one took n
bits and wanted to decide what they were, adding some of the information about
the bits, i.e. less than n bits, for example what n-1 of the bits were, to the
system T as axioms would not enable you to prove in the new system T' what all n
bits were. You had to add at least n bits. In more wordy terms, this means that
each bit of omega is proof-theoretically independent of all the others. Add it
to a theory which cannot prove what any of the bits of omega are and you could
still only prove what one bit of omega was.
Note that Matiyasevich's
solution of the Hilbert's tenth problem was similar, the only interesting
difference being that not every set of n bits of his real number were
independent, only some.
This is interesting mathematics but it's hardly
great mathematics, and Chaitin wants to be great, so he uses fantastical
language which is downright inaccurate at times. The bits of omega are not
random (omega is), and Chaitin's theorem for an individual bit amounts to
nothing other than "if T cannot prove what the bit is, then T cannot prove what
the bit is".
Chaitin's approach to research is stupid. His approach is
"if you can't solve a problem after working on it for a year, give up and guess
the answer, then move on to apply that answer to new problems".
This
philosophy has the following property. There is no positive integer N such that,
following Chaitin's philosophy, you will never lose more than N years worth of
mathematics due to that work being made based on a falsity. - Dan, 3rd December 2007
Before this becomes spam, a
quick point about different possible universes. First of all, It may be the case
that every true mathematical statement is realised in this universe.
Also,
"Hence, the different physical conditions in each universe
would result in different mathematical systems."
*If* there are
different possible universes then yes. But they are all mathematical models that
can be interpreted within mathematics. Mathematics shifts seamlessly from 3 to 4
to n dimensions, where n is arbitrary.
And to say a true mathematical
statement is true for any possible universe is surely not controversial, as we
define the term "possible" in terms of what is mathematically or logically
possible.
"To sum up, the Platonists' claim - that certain mathematical
structures which play dominant roles in our physical reality have an exalted
position in some Platonic realm - does not really stand up to investigation."
if only Platonists denied that n-valued logic had no application to our
universe. Even if you don't consider it as a logic, one could consider it as a
formal algebra that is bound to be of use somewhere. As far as I know,
Platonists only claim that 2-valued classical logic applies to mathematical
statements.
Not that I agree with Platonism. Talk of a mathematical
object existing is nonsensical. But it is also nonsensical to disagree with
nonsense. To ask whether 3/4 exists is nonsensical. To ask whether it exists in
the real numbers does make sense. That is my take on the Platonism/antiplatonism
debate. It is meaningless.
It is sometimes associated with Platonism the
idea that mathematical statements are either true or false. I think some, but
not all, mathematical statements are either true or false (I won't go into
that), for example a positive integer square can not be both a sum of two
squares and a difference of two squares. I verified that this was true myself, I
didn't "make" or "create" that it was the case, I merely verified that it was
so.
- Dan, 3rd December
2007
Thanks a lot for that Dan. Very
interesting. There must be something about this subject makes people want to
write essays!
First you said: "You keep saying that Math (thinking about
and studying mathematical concepts) was designed to describe reality". No,
that's not what I said. There's no "design" involved. "Design" implies choice.
We have no choice about the design of maths to model physical reality. Our only
design choices come in developing formal systems (discussed in the main text),
consistent mathematical systems which have NO match in physical reality. But we
have no design choices in the maths which we use in physics. As I said in the
main text, physics and maths are both axiomatic systems, so the human-led
development of maths will inevitably match the development of physics - no
"design" involved. But whether or not we develop those tools, physical reality
will carry on the same regardless. So physical reality seems more fundamental to
my mind.
And because there are no concious design choices involved,
mathematical models can develop in the unconcious mind in the womb, as you
suggest. Indeed, evolution is shaped by physical processes, so any mathematical
models produced in the human subconcious will no doubt match physical reality
(counting, as you suggest, matches physical reality in a way that would help
survival in the wild). I've never heard of a baby developing a mathematical
formal system, for example!
On a different subject, I'm not saying
Godel's work was not brilliant. I'm just saying it appears to have no match or
application in the physical world. It doesn't even appear to be used in pure
mathematics. It seems to be a mathematical dead end.
I think you make a
good point about mathematics being a constant in different universes, and your
Platonism comments. You know your stuff and write clearly. Thanks again. -
Andrew Thomas, 4th December
2007
"From the very start,
mathematics was designed to match physics, to be an effective tool."
That's the quote I noticed and took exception to, I equivocated "match
physics" and "describe reality". Whereas, I would agree with something like "The
situation of living organisms in the physical world and the laws of evolution
caused the developement of innate mathematical intelligence in these organisms.
These organisms then evolved into humans, and humans have since practised what
we call Math and Physics, and the study of each has informed the other."
We do develop formal systems, but remember formal axiomatic systems
(FAS) were first conceived in 1900 or so, and Math (and Physics) were around
long before that. Also Math is not a FAS, the whole point of Godel's theorem is
that there is no FAS which can express in it's formal language, and then prove,
every true statement about the nonnegative integers. In a sense it says that no
formal approach can tie down exactly what a nonnegative integers is. It would be
presumptuous to assume that the laws of physics can be accounted for by a FAS,
given that the laws of mathematics cannot.
As for what would happen
without us, there would be no Math or Physics. There would still be physics,
where physics is what we study in Physics, the laws of physics etc. A slight
linguistic distinction to make things clear.
And math, where math is
what we study in Math, equations, statements and numbers and stuff, it would be
math. One cannot say it would change, but also one cannot say it would remain
the same. Notions of changing and remaining the same do not apply to math,
because the things of math are not physical. Change and remaining the same
requires a notion of time, and time only applies to the physical. So all one can
say of math is that it would be math. One can equally not say it would exist or
would not exist, for that also would not make sense as I may have explained in a
previous post (the bit about 3/4).
As for Godel's theorem, Math is a
vast subject, and one would not expect to see a given theorem in a high
percentage of articles. It can be used in Math to prove that there is no general
algorithm for computing whether a Diophantine equation has a solution or not.
Although it is a statement about nonnegative integers, it is more of interest
philosophically than it is in the theory of numbers.
Godel's theorem and
it's proof did have physical applications in the developement of computers,
which are quite useful. An application to Physics would have to be
philosophical. One could say that if any question of whether some Diophantine
equation has a solution is of physical relevance, then the laws of physics
cannot be accounted for algorithmically (i.e. are infinitely complex).
Personally I believe that the laws of physics cannot be so accounted for, as it
is much more exciting, and nature never ceases to be exciting, or so I believe.
- Dan, 6th December 2007
Thanks for another great contribution, Dan. "Mathematics was designed to match physics, to be an effective tool" - oh, I did say that, didn't I! - Andrew Thomas, 6th December 2007