(The Tegmark paper on “The Mathematical Universe” can be found on arxiv.org.)

Does the theory warrant further scrutiny?

First of all the question was raised of what could possibly be gained by the approach favored by Max Tegmark; is it only a waste of time with no empirical consequences?

I would like to reframe the question: What does it actually mean calling reality mathematical?

1. An issue well pointed out by Max himself is maximal independence from word fluff: only relationships made explicit in formalisms, computations or structures constitute knowledge. This “making explicit” is a first boon, because when we speak in “normal” words everybody has different connotations and associations bundled with them; only by speaking in mathematics can we ensure that everybody associates the same meaning with what is said. Words are hidden inferences:
   

   Your brain doesn’t treat words as logical definitions with no empirical consequences, and so neither should you. The mere act of creating a word can cause your mind to allocate a category, and thereby trigger unconscious inferences of similarity. Or block inferences of similarity; if I create two labels I can get your mind to allocate two categories. Notice how I said “you” and “your brain” as if they were different things?

   Making errors about the inside of your head doesn’t change what’s there; otherwise Aristotle would have died when he concluded that the brain was an organ for cooling the blood. Philosophical mistakes usually don’t interfere with blink-of-an-eye perceptual inferences.

   But philosophical mistakes can severely mess up the deliberate thinking processes that we use to try to correct our first impressions. If you believe that you can “define a word any way you like”, without realizing that your brain goes on categorizing without your conscious oversight, then you won’t take the effort to choose your definitions wisely.

   In mathematics, we choose our “definitions” (rather: axioms, inference rules etc) very wisely – otherwise contradiction rears it’s ugly head. Everyday language use, on the other hand, is quite indifferent to contradiction (that is, by the way, the secret of politics ).

2. By adopting the Mathematical Universe stance, it is immediately obvious that everything we see emerges from an inside view. The “outside view” knows neither time nor space (as the concepts have meaning only in relation to structures on the inside). Think of this as a radically more extreme version of the block universe; have a look at endophysics – it is only a short wikipedia article but with nice references included. It should be obvious that a TOE (Theory of Everyting for non-physicist readers of the blog) we will develop must be endophysical – after all, we are inside the universe, and a TOE should also explain how our impressions arise from being inside this universe. Physicists usually call a TOE a theory which unites all four forces (electromagnetism, strong nuclear, weak nuclear and gravity); but I think we should adopt a broader view of TOE; maybe we should differentiate between a proximate TOE (having the usual meaning used in physics circles) and an ultimate TOE – bringing all knowledge into a coherent whole. Max’s Mathematical Universe would be an ultimate TOE.
3. The Mathematical Universe addresses the conundrum raised by many a thinker, most notably be Eugene Wigner: The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Everybody who denies the mathematical nature of reality is welcome to present more plausible alternatives for the effectiveness of mathematics. In this sense, even if the Mathematical Universe Hypothesis were wrong, it would be a good catalyst for further philosophical inquiry.
4. An important point which can’t be stressed enough because it is so deeply ingrained in our thinking: from the Mathematical Universe viewpoint immediately follows the elimination of essences. Most traditional Western philosophy is concerned with the “essence” of an object/subject; or, similarly, is occupied with categorizing things in ontological hierarchies. These approaches are largely unsuccessful and thus have lead to postmodernism and relativism. The Mathematical Universe immediately shows why these approaches have failed in the last 2500 years: because there are no essences apart from relations.Even if you are not a philosopher and have not heard of essences: if you were brought up in a Western context, it is pretty much guaranteed that you are thinking in essences; it is part of our cultural background; it shapes the way we categorize your knowledge. See Eli’s post How an Algorithm Feels From Inside for how this essence feeling arises.This predilection for essences is of course a human phenomenon, arising from our brain architecture – it applies to all people. Why do I stress a Western context then? Because Buddhists/Taoists/Zen-philosophers have, through long meditation, seen through this “mind trick” – mind as seen from the inside. This does not mean that every chinese/japanese guy you’ll encounter will have understood this: after all, the Zen masters are revered in the East, because it is so difficult to dissolve these concepts.But in the East, the dissolution of essence exists as a cultural background, ready to draw on, whereas in the West it is only a minority position not vigorously pursued.
5. So we have moved away from Aristotelian essences. But the Mathematical Universe is not traditional Platonism, which speaks of ideals and mere shadows and denies reality perfection; to quote from the Wikipedia article on Platonism:
   

   The central concept is the Theory of forms. The only true being is founded upon the forms, the eternal, unchangeable, perfect types, of which particular objects of sense are imperfect copies. The multitude of objects of sense, being involved in perpetual change, are thereby deprived of all genuine existence.

   The Mathematical Universe is in this sense very contrary to traditional Platonism: it does not say that existence is a mere shadow, an imperfect copy of some eternal object “out there” in some inaccessible realm, but in fact the platonic relations are all there is. No shadows, no caves, no torches. Only ideal structures. The moment you are currently experiencing is encoded is this Platonic mindscape. I dearly recommend reading Julian Barbour’s The End of Time and his concept of time capsules.

So, the Mathematical Universe concept promises to merge the oldest of dichotomies in Western philosophy: Aristotelianism versus Platonism. No small feat, if you ask me.

Above, we have seen that the idea of the Mathematical Universe shows much promise; so it warrants turning one’s attention to it.

Addressing some criticism

1. Bee posits The Principle of Finite Imagination; which I like very much; it was one of my own initial reactions in my first encounter with the Mathematical Universe idea; I talked with Max Tegmark in the aftermath of this conference, raising much the same issue. Bee thinks of a “Level 5: Beyond Mathematics”; but I would just call it advanced mathematics, maybe not even recognizable to us humans as mathematics yet; think of future AIs uncovering exquisite structures we have not thought
   of yet, maybe are not even capable thinking of with our little human
   brains.

   I am perfectly fine with this non-human mathematics, as I do not think that humans are the evolutionary maximum of all possible epistemic agents. But Max does not mean current mathematics, he means all possible mathematics; I think what Bee means with Level 5 is contained in Level 4 already (all mathematically possible structures).

   My other concern was: what about a universe without any structure at all? Max said that he thought that this would simply correspond to the empty set, and after thinking long and hard about this I have come to the same conclusion (in hindsight it may be obvious, trivial: but we all have our mental models of the world and some parts readjust more slowly than others; that was my personal “barrier” ).

2. Those who think the idea is too far out (Max says this is even a bonus of the theory, and I agree – why should reality correspond to our intuitions developed in a provincial evolutionary context?) should be advised of the development in philosophy of science in the scientific realism debate: structural realism. To keep things short I will only quote the first sentence from the SEP article:
   

   Structural realism is considered by many realists and antirealists alike as the most defensible form of scientific realism.

   Max’s version corresponds to the ontic variant of structural realism. But people who adhere to epistemic structural realism should think long an hard: if indeed structure is the only thing that can be known, what additional thing is there to talk about? The ontic variant is the most parsimonious version of structural realism, and in metaphysics above all we should be parsimonious, lest it degrade into mere fiction.

3. And now for the most interesting objection raised by Bee and a colleague of hers:
   

   Scientific arguments aside, my reason to not believe all of reality is maths is that for me the interesting thing about maths is not that we are able to use it. The interesting thing is what Plato above called ‘an intuitive leap’. Call it intuition, a believe, a hope, or a conjecture. The interesting thing for me is the capability of the human mind to observe, and to translate this observation into something more general, taking away clutter, finding the patters, playing around with them. It is the process of this translation that I find important, not the result, the language in which we formulate it.

   I think this is essentially also the question Olaf Dreyer has been asking in your talks here at PI, I think he asked a similar question in both talks, and I think you misunderstood the question both times. He was asking how come that we connect much more with the ‘mathematical structures’ on our notepads than the actual symbols contain. How come we are able to get an intuition for what these things ‘do’, that what makes the essential difference between mathematical proof and physics, the intuition advantage that physicists can bring into mathematics. Where does this come from? Can we ever describe this by a mathematical equation?

   First of all, one must be careful not to confuse the modes of “being” and “knowing”. “Knowing” is a subset of being; there is being without knowing (that is Max’s External Reality Hypothesis), but no knowing without being (a contradiction in terms/concepts).

   When we think about mathematical structures, make intuitive leaps etc, this is a mode of being (encoded in the Mathematical Universe); but when we make the connection to reality (physics) this is a mode of knowing: a mathematical structure (knower) reflects parts of his local mathematical surroundings (world) – this is then called knowledge, and is together again encoded in the “bird” structure.

   Notice how the Mathematical Universe raises interesting research questions: why do some structures find themselves in the proximity of others etc? (I can’t help from feeling very excited by these prospects; the Mathematical Universe idea seems to give a first handle on truly fundamental questions.) I will elaborate on this being/knowing distinction and how they can be visualized in the Mathematical Universe in future posts.

   To clarify, I repeat the above phrased a bit differently: Thoughts are reflections of mathematical structures. Thoughts which do not correspond to our immediate reality may correspond to other parts of modal reality, inaccessible from here, but no less real for that (this will have beautiful consequences when extrapolating further -> see a future post).

   We have the power to reflect on many structures: that is why mathematics is about ultimate creativity: it is exploration of all possibilities. Indra’s net is a good intuition pump. Maybe we humans can’t yet explore them all (Bee’s “Level 5″) but by further evolving, constructing AIs etc we will increase the accessible mindscape.

   To answer Bee’s and Olaf’s question directly: the ability of humans to play with ideas, connect with reality, translate and transform etc is a mode of being in the mathematical universe; when it is well done, that is, mathematical structures are successfully reflected in the cognitive system, the result is knowledge. Inaccurate reflections are dreams.

4. Bee also raises the question on how our thoughts could be mathematical structures when they are inconsistent, wrong or undefined? The answer is related to my explication above.”Undefined”, “illogical” thoughts happen when you reflect mathematical structures incompletely. An example: say you view an object through a mirror, but part of it is cut off: your “undefined” or “inconsistent” mathematical thought would be the system:
5. Are we leaving the domain of science with all this (this is a valid criticism!)? Depends on how you think scientific knowledge is defined (ie falsificationism vs confirmation theory).But even if we are leaving the precints of hard fact and the core of the natural sciences: certainly we are not leaving the domain of philosophy. And philosophy is about making sense of the world. And good philosophy is philosophy which does not contradict scientific evidence, on the contrary, which is even supported by scientific evidence. I have made a little diagram to make my point (it is not intended to be in any way canonical, only illustrative):
   

   Circle of Knowledge

   Here the texts for people reading with a text browser (the image shows outward moving circles representing different levels of scientific certainty (A) to (D) and three circles apart representing unscientific positions (E) – (G) ):(A) Hard empirical facts.
   (B) Less certain empirical facts.
   (C) Logical deductions from empirically successful theories.
   (D) Plausible reasoning using the same assumptions that lead to empirically successful theories.

   (E) Implausible reasoning.
   (F) Bad reasoning.
   (G) No reasoning.

   The Mathematical Universe Hypothesis is located roughly in the (C)/(D) section, supported by evidence from (A) and (B); as we move outward from the inner core (A) things get less and less sure. But we should not refrain from reasoning especially in the (D) section, because it is indeed here that new ideas for unification in science or even new experiments arise. (D) is the fecund field for discovering new knowledge.

   The problem is that most people who don’t agree with the Mathematical Universe don’t argue from (A), (B), or (C); not even from (D) actually.

   Rather criticism is raised from (E), (F) or (G), because of philosophical or personal considerations. So I would urge to constructively criticize, coming from (D).

A mathematician’s and a physicist’s dilemma

In the end, the question for critics remains: how do physical brains of mathematicians come to know about acausal, atemporal abstract objects (as the traditional Platonist view will have mathematical objects)?
(BTW, If you still believe that thoughts are independent of physical brains I recommend this entertaining book: Oliver Sacks: The Man who mistook his wife for a hat; or Phantoms in the Brain by V.S. Ramachandran; you should seriously reconsider any dualisms).

The immediate objection which comes to mind is that one could adopt another philosophy of mathematics (not Platonism) and then simply regard the success of math as successful pattern matching. That is the way I took first; but, of course, it does not work. The way leads wonderfully back to the Mathematical Universe; but that is for a future post.

The physicicst is actually in an even worse situation: he/she is using a tool which “magically” works and he doesn’t even know why: no, she doesn’t even think about it: why mathematics works is a metaphysical question, not science, and therefor best not thought about (or so the argumentation goes)!

Conclusion

Why should we ignore knowledge? That mathematics works to describe reality is an empirically confirmed fact! Use this knowledge, do not ignore it! This is indeed one of the major principles of rationality: never ignore knowledge!

Let evidence guide our beliefs, and not a priori beliefs (=childhood beliefs, cultural background) guide the way we weigh evidence.

I have hinted at some future posts above, where I hope to clarify some issues which may have remained dark or only hinted at in this first outline.

But enough for today.

Technorati Tags: mathematical universe

Technorati Tags: mathematics, phd, philosophy, philosophy of science, physics

For the philosophical relevancy, this quote is a nice demonstration:

Jerry Bona once said,

The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn’s Lemma?

This is a joke. In the setting of ordinary set theory, all three of those principles are mathematically equivalent — i.e., if we assume any one of those principles, we can use it to prove the other two. However, human intuition does not always follow what is mathematically correct. The Axiom of Choice agrees with the intuition of most mathematicians; the Well Ordering Principle is contrary to the intuition of most mathematicians; and Zorn’s Lemma is so complicated that most mathematicians are not able to form any intuitive opinion about it.

Axiom of Choice

Also be sure to read a bit about the Banach-Tarski Paradox, which shows that AC can lead to problems when applying theories which use AC to problems of physics. (Of course, this is not a practical problem because to my knowledge most (all?) of the math used for physics can also be expressed without AC, although in a more complicated exposition.)

But one conclusion can be drawn: one must be very careful in introducing (uncountable) infinities an also when reasoning with infinities. Our intuitions have evolved and are learned in a finite, albeit vast world (also in space, we can’t see past the Hubble volume).

Technorati Tags: mathematics, philosophy of science

Overcoming Bias: The Simple Math of Everything

I called it a concept hierarchy of science; and mathematics is nothing else than concepts in relation. (the uninterpreted equations are only meaningless symbols). So, the equation plus the mapping to the relevant concepts in the science would constitute a “core idea” – maybe the Overcoming Bias community has the momentum to start a concept hierarchy?

Technorati Tags: mathematics, philosophy, philosophy of science

How the Brain Maps Symbols to Numbers: Scientific American

Technorati Tags: cognitive science, mathematics

Outside In

Technorati Tags: mathematics

How should logarithms be taught?

Be sure to read the comments, the discussion get’s really interesting there!

Technorati Tags: mathematics

Re: [FRIAM] math and the mother church

I can recommend the FRIAM list very much – always interesting discussions going on there.

Technorati Tags: mathematics, philosophy of science

Good Math, Bad Math : Order From Chaos Using Graphs: Ramsey Theory

Technorati Tags: complexity, mathematics, philosophy of science

Independent means that CH can neither be proved true nor false within the ZFC axioms.

Here is a nice blog post explicating a little more:
EvolutionBlog : Paul Cohen Dead at 72

http://scienceblogs.com/mt/pings/37198

Technorati Tags: mathematics

In 1956, Maurits Cornelis Escher completed a drawing called “Print Gallery”. The drawing depicts a young man looking at a print in a gallery that is deformed almost beyond recognition. There is an enigmatic white area in the center of the image.

In 2003, a group of mathematicians at Leiden University, led by Prof. Hendrik Lenstra succeeded in unraveling the mathematical structure of the image. Once this structure was known, they could “complete” the image by filling in the famous white spot with the help of a computer algorithm.

A summary of their method was published in a paper, and received a lot of acclaim, not only in academic circles, but also in the general press.

Technorati Tags: fractals, mathematics