Below there is a problem that I have formulated in order to understand better what is mathematics. There are two related dicussions:
I will start first with the final version of the problem (Bruno’s comments were very helpful) and after that I will present my messages that led to the problem and the final version of it.
An experiment to perform in order to find experimentally what is the meaning of Pi under the physicalism hypothesis
I assume physicalism. From SEP
“Physicalism is the thesis that everything is physical, or as contemporary philosophers sometimes put it, that everything supervenes on, or is necessitated by, the physical.”
“The general idea is that the nature of the actual world (i.e. the universe and everything in it) conforms to a certain condition, the condition of being physical. Of course, physicalists don’t deny that the world might contain many items that at first glance don’t seem physical — items of a biological, or psychological, or moral, or social nature. But they insist nevertheless that at the end of the day such items are either physical or supervene on the physical.”
“Physicalism is sometimes known as ‘materialism’; indeed, on one strand to contemporary usage, the terms ‘physicalism’ and ‘materialism’ are interchangeable.”
The Pi number enjoys extensive use in physics. This raises the question what Pi means under the physicalism hypothesis.
Below there is a description of the experiment that one could think of to check the relationships between Pi and physicalism.
Let us take a completely isolated bunker where the experiment begins. There are two mathematicians in the bunker and the initial conditions are enough so that mathematicians can comfortably work for awhile and prove the existence of Pi on a paper. Eventually the oxygen in the bunker will run over and both mathematicians die.
From a physicalism viewpoint, we have a dynamical system that eventually comes to the equilibrium state. Because of experimental settings, we can neglect the interaction with environment and I hope that this could be done even for the quantum mechanics treatment.
The experiment takes an operational approach to what Pi means. During the initial stage of the experiment mathematicians prove the existence of Pi. This should be enough to claim that Pi is present in the bunker at least for some moments.
Questions to discuss
How Pi supervenes to the physical states of the bunker with mathematicians?
Is Pi invariant in respect to states of the dynamical system in question or not?
Quotes from my messages to the embryophysics and everything lists
03.03.2012 17:32 to embryophysics:
Yet, it is unclear what happens with mathematics. Is this a thing independent from an intelligent mind or not?
This first concern physical laws, because if mathematics is dependent on a mind, it is unclear how physical laws has guided the matter when there was no mind.
This also concerns the discussion about hardware and software. When we speak about software, I guess, we mean an algorithm. The latter is a pure mathematical construct and it is unwise to search for it in the Nature. We may find some physical system that implements some algorithm, but in my view this is not exactly what the term algorithm refers to. Again, if mathematics is mind dependent then there were no algorithms (software) in the Nature when there was no mind.
03.03.2012 22:42 to embryophysics:
It is still hard to understand what information in biology is. You say that is an important concept but you do not say what it is.
I agree that information is an idea, human abstraction. On the other hand, I do not understand how chemistry could develop such an idea. In order to explain this better, I would suggest following. Let us consider a situation where two mathematicians talks with each other about pi. I would agree that after all this is just a bunch of molecules even with very complicated autocatalitic reactions. What I do not understand how the number pi appears from that chemistry.
04.03.2012 10:13 to embryophysics:
Recently I have learned what biosemiotics is. By the way in his paper
Barbieri, M. (2007). Is the cell a semiotic system? In: Introduction to Biosemiotics: The New Biological Synthesis. Eds.: M. Barbieri, Springer: 179-208.
Barbieri proves that the cell is also a semiotic system. He has found the way on how to play with definitions in order to find meaning during work of DNA/RNA/proteins. His paper is quite logical though, he gives definitions and then applies them. It is more easier to follow his logic in the paper as the logic of using the term information in the current discussion. By the way his conclusion about information at the DNA level is quite different.
I doubt however, that this will help us to understand how the number pi appears when we consider two mathematicians at the chemistry level.
04.03.2012 13:27 to everything list:
An experiment to perform in order to prove experimentally whether Pi exists independently from the mind
The idea came during discussion on embryophysics list
Below there is a description of the experiment that one could think of to check the relationships between Mathematics, Mind and Nature (the MMN experiment). In my view this could be done as a real experiment (so this is actually not a thought experiment) provided we find two mathematicians who agree to sacrifice their life for science. I believe that this should be not that difficult provided the importance of the experiment for the modern science.
Let us take a completely isolated bunker where the experiment begins. The initial conditions are enough so that mathematicians can comfortably chat for awhile with each other about Pi and prove that it exists. Eventually the oxygen in the bunker will run over and both mathematicians die. From a viewpoint of a natural science, we have a dynamical system that eventually comes to the equilibrium state. I assume that at the beginning when mathematicians prove that Pi exists we have a consequence of physical states where Pi exists indeed. If you are in doubt, please suggest any other physical states where you say that Pi exists. The goal of the experiment is to establish what happens with Pi at the end when the system reaches the stationary state.
Because of experimental settings, we can neglect the interaction with environment and I hope that this could be done even for the quantum mechanics treatment.
Before the experiment will be perform in real, you can take your bet on whether Pi is retained after the death of mathematicians or not.
04.03.2012 17:12 to everything list:
Actually it is not a joke. I guess it is my first step toward Platonia. As I am a chemist by background, the problem might be not mathematically correct indeed. Yet, if you could help, we could improve it in this respect.
The background is as follows. I am a chemist and I am still at the level of what you refer to as physicalism or mechanism. Before I consider your theorem, first I would like to understand better in my own terms what physicalsim and mechanism mean and what are the limits. When you talk about this, it is too fast for me.
According to a common view in natural sciences, a human being (and hence mind) has been created during evolution. Right now however, after following discussion here, I have a problem with mathematics along this way. Science has been pretty successful with mathematical models in physics, chemistry and even in biology. Yet, according to my current view, mathematics has been created by the mankind. Thereafter I have got suddenly a question, why mathematical models (physical laws) are working at all to describe the Universe when there was no mind. The mathematics, it seems, was not there at the times of Big Bang.
We cannot repeat Big Bang to understand this. According to the current economic situation, it is highly unlikely that taxpayers are ready to spend money on bigger and bigger particle accelerators. Hence my proposal. If we cannot repeat Big Bang, then for a relatively small budget we could make easily a local heat death of a small Universe with two mathematicians and see what happens with mathematics there. In a way, we repeat evolution in the reverse direction.
It would be nice to exclude mind out of consideration at all but as this is impossible my goal was to reduce its role as possible. We know that mathematics is what mathematicians do. Pi is a nice number and most of taxpayers have heard about it. In the experiment we could allow mathematicians to write the prove that Pi exists on a paper, it would be even simpler. If you think that some other mathematical object would be nicer, please make your suggestion.
So, at the beginning of the experiment we have mind (two working brains of mathematicians) and they prove on the paper that a given mathematical object exists. An open question to discuss is what happens with this mathematical object at the end of the experiment.
04.03.2012 21:07 to everything list:
I understand your logic but then immediately comes a question where mathematics objects exist. In this case Bruno is consistent when he says that everything is formed from the mathematical objects in Platonia. Do you mean the same?
I personally still at the position that there are some material objects, atoms, molecules, crystals, etc., that are independent from the mind. I believe that this is quite a typical position for natural sciences. Then it is hard to imagine how mathematical objects coexist with physical objects. Some sort of dualism?
05.03.2012 10:06 to embryophysics:
We are at the point now where it is good to be back to definitions.
What I observe in your experiment is as follows. When scientists develop a DNA segment for gene therapy, they do use information, no doubt. When however the DNA molecule causes some reactions in cells, I do not see any reasonable definition that could distinguish between the DNA and other molecules in term of information.
If you know how to define information so that only DNA molecule possesses information and other simple molecules do not, please disclose your definition (it could be incomplete and informal, this should be okay).
In my personal view, information is just a mathematical concept, similar to algorithm, or even to the Pi number. Pi enjoys extensive use in science (also in biology), so what? Could we find Pi in Nature? What is the difference between information and Pi in this respect?
I see a general problem in my position, as whether mathematical objects are dependent or independent from mind. Many mathematicians claim that they are independent (and reside in Platonia). Yet, I should say that the idea that Pi lives in Platonia is as crazy, as that information lives in the DNA.
I have suggested an experiment on the everything list to check if Platonia exists
Two Mathematicians in a Bunker and Existence of Pi
We can use it the same way to check if information exists in Nature, just replace Pi by information there.
05.03.2012 12:23 to embryophysics:
I like a lot this quote about physicalism from SEP
“The first thing to say when considering the truth of physicalism is that we live in an overwhelmingly physicalist or materialist intellectual culture. The result is that, as things currently stand, the standards of argumentation required to persuade someone of the truth of physicalism are much lower than the standards required to persuade someone of its negation. (The point here is a perfectly general one: if you already believe or want something to be true, you are likely to accept fairly low standards of argumentation for its truth.)“
I should confess that it describes my personal feeling very well. Cheers to philosophers.
05.03.2012 16:25 to embryophysics:
A good question and it has been discussed extensively by philosophers for already thousands years. I right now listen to lectures of Prof Hoenen on philosophy, for example
and I should say it is very enjoyable. In any case, it is important to distinguish between an object sun and the term sun. When we speak of the term then there are two potential meanings, first just a name, Sun as a notation for a particular star, and then a term that describes any sun (in this particular case, it is better to use the term star). Probably you would agree that both the name Sun and the term star are mind dependent.
By the way, not all philosophers agree with that. For example Gottlob Frege would put the term star in Das dritte Reich (has nothing to do with Hitler).
I guess that you have meant an object and here you will find different opinions. For natural sciences nowadays the most common viewpoint is materialism and I have made a link to physicalism where materialism is defined more formally through physics. I think that this is exactly in what I believe right now and probably you keep this position as well.
Now according to physicalism the object Sun is mind independent. So far so good. The question is however what Pi means according to physicalism/materialsm. Thereafter my experiment to better understand it.