- Mathematics & Mathematicians
- Mathematics & Models in Science
- P-adic numbers
Question and Answer;
That is the format of this first exploration of Matti’s work.
What Matti found, if formulated in his own form of mathematics.
But Matti could find this, because he formed his own form of mathematics.
So, mathematics stands central.
It is at the same time the instrument for thinking, and recording.
Does mathematics really bridge the ideas in the mind and words on paper?
Is the mathematics of/for thinking, the same as the mathematics of formulation/formulae?
I started with the following questions to Matti; this was my email:
O#o van Nieuwenhuijze, MSc, MD kirjoitti 24.05.2012 kello 00:41:
Thx, I think it is great, what is possible in this series of interviews!
For tomorrow (in your time zone, today already) it might be nice to talk about mathematics.
Yet, rather, in such a way that anybody can sense, feel, grasp what mathematics is about.
Few people remember that it is a symbol(ic) language, which WE create.
It may be wise to speak about the feeling, “the beauty”..., of mathematics.
Later, we will be able to get into logic, formulations and formalisms, and calculation/results. But first..
- What is mathematics about, and a language for communication and ... Thinking?
- Why are there so many different forms of mathematics; what makes them different?
- You discovered/created/designed/use different models of mathematics; why, how?
- What are the specific models of mathematics which you now use, and why?
This will probably give a good general starting point.
As before, it starts with the intuitive/feeling level.
It then goes into practical choices and uses.
Later we can look, in detail, in the result that offers.
You already mentioned “sub-manifolds” (systems), “quantum” (dimensional operations) and “p-adic numbers” (calculus); maybe the most central of these can be of help as example to illustrate how you interprete/use mathematics?
1) what is mathematics about, and a language for communication and ...Thinking?
2) why are there so many different forms of mathematics; what makes them different?
I have not thought mathematics at this meta level.
My view is rather mathematics centered.
I indeed realized yesterday that the core of TGD is about new ontology: about what exists.
1) What is ontological status of mathematical structures, theories?
Do they exist in some sense? Do they exists like matter exists?
[Poposal: theory about physics is the physics. The objective realities that we call quantum states are just their mathematical descriptions. Here my view is similar to Tegmark's except that he does not try to identify the mathematician as conscious entity. Mathematical consciousness form quantum jump between different quantum states, mathematical objects. Travelling around in Platonia;-)]
Yes this reveals to me my view;-): mathematics is not a mere tool of thinking but the very objective existence.
2) Feedback aspect of mathematical consciousness is overall important:
seeing ones thoughts is certainly fundamental for developing mathematical consciousness.
3) There are ontological questions also about status of Boolean logic. Is it part of quantum existence?
[Fermionic Fock states as representations of Boolean statements and zero energy states in zero energy ontology (again a little bit of ontology!) as statements of type A-->B or more precisely, their quantum superpositions: this is what thinking mostly is rules, quantum states as quantum rules stating laws of physics!]
4) What is the role of p-adic mathematics in understanding mathematics?
[p-Adic topology appears in the context of so called Stone spaces.
One of them is the space of possibly infinitely long bit sequences.
It consists of 2-adic integers forming continuum. p>2-adic numbers seem to related to p-valued logic or effectively p-valued logic.
Therefore identification as geometric correlates for cognition.
p-Adic surfaces as thought bubbles. Descartes.
p-Adic classical physics is non-deterministic and this inspires also description of intentions p-adically: intention but not yet deed as p-adic space-time surface.]
5) Self reference is one fundamental aspect of mathematics. Writing a formula expressing mathematical thought is self reference, seeing own thought from above. Climbing to a new level in the hierarchy of existences. What is this hierarchy?
Does self reference reveal itself in the very structure of mathematics - and physics if physical object is theory about itself?
[Here the hierarchy of infinite primes pops up.
Maybe it is too abstract and technical. Second hierarchy is quite recent and relates to the generalization of arithmetics.
Numbers and + and * are replaced with Hilbert spaces with direct sum and tensor product and define shadows of deeper physical existence. Even more: points of Hilbert spaces are represented by numbers which can be replaced by Hilbert spaces, and this replacement can be done again and again! Self reference! Maybe here is the proper manner to describe self reference. Points are not structureless but have fractal structure, something speculated also by Grothendieck. Even space-time points would evolve in evolution.
There are intelligent and less intelligent space-time points! This would mean taking number theoretical Brahman=Atman -or algebraic holography, to its extreme. Single space-time has infinitely complex number theoretic anatomy able to represent entire universe! ]
Your second question is very general. As such I am not able to say much about it. There are however two basic manners to think mathematically. Geometric-topological based on visualization and algebraic based on highly symbolic language. They are very different moods of mathematical thinking. About this I am able to say something.
[Physics as infinite-D geometry for the world of classical worlds as first mode of TGD. Physics as generalized number theory as second mode of TGD.]
A brief note about vortices and magnetic flux quanta. As I explained, the purely local mathematics of magnetic field is that of incompressible fluid flow. Water is the fundamental example. Vortices have their magnetic counterparts.
What about global mathematics of magnetic fields? Here TGD differs from Maxwellian vision which is linear. In TGD fields are geometrized in terms of sub-manifold geometry. Once you know space-time as 4-surface in 8-D space you know all the classical fields.
A new phenomenon is flux quantization which is observed also for ordinary hydrodynamical vortices in super-fluidity. In TGD framework this quantization is quantization of space-time. For instance the magnetic field decomposes into flux quanta, spacer-tme sheets which are typically flux tubes or flux sheets having typically finite size in some direction(s). This means that physical systems have field identity, magnetic body.
The braiding of magnetic flux tubes involves knotting and linking and would be a fundamental phenomenon responsible for the most fascinating aspect of biological information processing. Idea would be simple: connect molecules by flux tubes. When they move around these flux tubes get braided and code the motion to a memory. These braiding patterns would be also crucial in making possible quantum computation like activities.
Example: lipid layers of cells are liquid crystals. 2-D liquids. The flow of this liquids has vortices and these vortices can coded to memories if one connects lipids by flux tubes to say DNA. This is like connecting dancers to the wall by threads and storing the dynamics of dance to the braiding of threads!
Names of mathematicians
One of the difficulties in dealing with mathematics, and mathematicians, is what i call ‘name dropping’. “Descartes”. “Newton”. “Hilbert”. And so on.
These are the name of mathematicians, but in mathematics they do not represent their name. They do not represent them. But they represent their thinking; or rather: their way of thinking. But only in the manner, or in the extent, of the way the mathematician thinks of that, when he refers to that.
Flavours of thoughts
It is like someone speaking of the memory of the flavour of a cheese that they ate. For them it is fully meaningful.
Yet, that meaning is significant only within their development/experience, and only with respect to what they try to communicate.
It is as if they call up the memory of the flavour of that cheese in making a recipe for a meal that they might like to cook up.
For me (O#o) “Mathematics is the score for a choreography for the dance of ideas in our mind.
Like a musical score, which defines how instruments are played in an orchestra.
Yet, every instrument/player/idea and thus every performance/concert/conclusion will be different and unique; determined by context conditions.
The mathematician is not a pianola; the mathematical formula has different effects in every mathematician.
Therein, ‘methinks’, mathematics is no different than language.
A formula is the equivalent of a word.
The meaning of the word/formula depends on how it is spoken.
Yet, the meaning is meaningful only as expression of the underlying ideas.
2) Mathematics & Mathematicians
How Matti learned to use mathematics; it worked - no need for understanding.
Then it became clear that those models caused the conflicts in physics (symmetry versus broken symmetry).
By using different mathematical models, the break could be healed.
(In-between reflection: the different forms of science are consequence of different mathematical models)
3) Mathematics & Models in Science
Instead of using four mathematical models for for different forms of science (classical, relativistic, quantum and unified field), Matti uses his same one m model in four different interpretations.
4) P-adic numbers
What they are and how they work? It does not seem to matter: Matti uses them and is able to come up with much more simple understanding of much more complex processes in physics.