Discuss:Science, Engineering, Knowledge:Foundations of Science
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Foundations of Science
henrik - Sun Mar 14 9:43:59 2004
Foundations of Science
At first, this was going to be a posting for the Axiomatic Method Forum. However, when I finished writing this posting I realized that applying the axiomatic method to science really deserved to be treated as topic of its own.
The axiomatic method is extremely powerful and educational in the sense that it allows us to derive so much knowledge from a small set of axioms and it makes our understanding of the issues very clear. If one understands the axioms and the rules of inference that we can use on these axioms then everything is crystal clear.
It is also the case that 'everyday mathematics', as practiced by your average professional mathematician, who studies, teaches and does research in topology, combinatorics, differential equations, complex analysis, etc can be formalized using our good old friend ZFC.
However, viewed from a certain perspective this might be unfulfilling. Mathematics in itself says nothing about which formalism is better to describe reality, nor does it make any attempt to do so. For the mathematician, investigating the consequences of Euclidean geometry might be as fulfilling as exploring the consequences on Non-Euclidean geometry. However, for the scientist or the inquisitive mind, who would like to know more about reality itself this is clearly not sufficient. It is important to refer back to reality to check which axioms seem to describe reality better. This is exactly what Sunny would like to do every time he says that we need to only choose axioms that are consistent with reality.
Using mathematics and referring back to reality to create better models of certain phenomena of the world is what we would traditionally call 'Science'. However, there is a big difference between choosing a set of axioms to describe reality and choosing a differential equation to represent a particular phenomenon in reality. Unfortunately, most scientific research is conducted in such a way as to make reliable predictions about certain phenomena using some mathematical formalism like a differential equation, regardless of whether the theory as whole is coherent and can be derived from a simple set of axioms that we consider to be true about the world.
What this means is that there is a big gap in our knowledge. On the one hand, mathematics has solid foundations by using the axiomatic method, but does not really care about the true nature of reality. On the other hand, natural sciences like physics care about the true nature of reality, but they lack an axiomatic foundation and hence are doomed to rely on a more or less scattered collection of mathematical equations that seem to describe their phenomena reasonably well.
There seems to be a very strong need for foundational research in the natural sciences. There is a need for an axiomatization of contemporary physics and other areas of science.
As far as I know, up until now nobody has been able to come up with a full axiomatization of contemporary physics. In fact, no one has even been able to fully axiomatize the much simpler theory of Newtonian Mechanics.
I do not claim that the ideas I present here are new. I am just calling attention to these issues, since not everybody is necessarily aware about these ideas. For example, Harvey Friedman discusses the foundations of physics on the Foundations Of Mathematics mailing list.
One can refer to:
Harvey Friedman [FOM] 206:On foundations of special relativistic kinematics 1
There is quite a bit of very recent and very interesting research in the direction of formalizing physics.
Here are a few references for those of you who might be interested.
[1] Andreka-Madarasz-Nemeti: Logical axiomatizations of space-time.
Samples from the literature.
http://www.math-inst.hu/pub/algebraic-logic/lstsamples.pdf
[2] Andreka-Madarasz-Nemeti: On the logical structure of relativity
theories. http://www.math-inst.hu/pub/algebraic-logic/PartI.pdf
[3] Andreka-Madarasz-Nemeti: Logical analysis of relativity theories.
http://www.math-inst.hu/pub/algebraic-logic/foundrel03nov.pdf
[4] Madarasz-Nemeti-Toke: Generalizing the logic-approach to space-time
towards general relativity: first steps (draft)
http://www.math-inst.hu/pub/algebraic-logic/loc-mnt04.pdf
--henrik Sun Mar 14 9:43:59 2004
joryea - Wed Aug 18 17:55:42 2004
Science and mathematics are completely connected, are they not? Geometry of the universe explained through algebraic formula's represented as a scientific hypotheses.. =) actuality within reality.
--joryea Wed Aug 18 17:55:42 2004
charmed_quark - Wed Aug 18 19:57:39 2004
An axiomatic basis for physics would be synonymous with a unified field theory, correct? I think the current state is that M-string theory and Quantum gravity provide models, but no observations have been made that confirm these models to be uniquely correct.
The next question would be, does this end science? When systems become sufficiently complex, an axiomatic approach is inadequate. The old line is that psychology is biology and biology is really chemistry and chemistry is really physics and physics is really math and math is really philosophy. It may be fundamentally true, it in practical terms specialization will still be necessary.
--charmed_quark Wed Aug 18 19:57:39 2004
henrik - Thu Aug 19 4:28:29 2004
"An axiomatic basis for physics would be synonymous with a unified field theory, correct?" charmed_quark
No, not exactly. They are related though in the following way...
Suppose that after years of painstaking research, we are convinced that M-string theory is the correct unified theory of everything, ie all the observable phenomena in physics can be predicted using this theory. Then, an axiomatic foundations for Physics would consist in formalizing M-string theory. In other words choosing some logical language and finding the right set of axioms that allows one to derive all of M-string theory only using the rules of logical inference on that small set of axioms.
We are probably quite far away from the goal of axiomatic foundations for Physics on both fronts of the inquiry. On the scientific side, M-string, Quantum Loop Gravity etc... still remain quite controversial and it is not clear if any of these alternatives is the correct theory to choose. On the formal logical side, logicians are still working on formalizing fragments of relativity theory and newtonian mechanics. Thus, there seems to be quite a long journey before we get axiomatic foundations for Physics.
Another reason it might take a long time to get Axiomatic Physics is that there is currently not much interest nor funding for projects to axiomatize scientific theories.
--henrik Thu Aug 19 4:28:29 2004
henrik - Thu Aug 19 4:55:35 2004
"The old line is that psychology is biology and biology is really chemistry and chemistry is really physics and physics is really math and math is really philosophy." charmed_quark
You are referring to Reductionism. The idea that Sociology can be reduced to Psychology, Psychology to Biology, Biology to Chemistry, Chemistry to Physics and Physics to Quantum Mechanics. Thus, in principle if we knew exactly how Quantum Mechanics worked and if we had a powerful enough computer we could make sociological predictions like who is going to win the next election.
Reductionism is a fascinating thought and is a source of inspiration for a lot interesting research. However, it can not be taken too literally, because even if God himself gave us the right set of equations to describe quantum mechanics, we wouldn't have the computing power to make macroscopic predictions. Moreover, even if we had a powerful enough computer to make mascropic predictions we wouldn't really understand what was going on. This is the "Emergent Phenomenon" property that many people have described (Wolfram for example). Even with a very small set rules you can create very complex and unexpected phenomena.
--henrik Thu Aug 19 4:55:35 2004
