[FOM] 206:On foundations of special relativistic kinematics 1

[Harvey Friedman]

In the posting Foundational Thinking 1, 1/20/04 1:24AM,

http://www.cs.nyu.edu/pipermail/fom/2004-January/007857.html

I referred to the great potential applicability of foundational thinking in
contexts far beyond the foundations of mathematics.

Of course, this is anything but a new idea. However, I do not believe that
an explicit and systematic attempt to apply foundational thinking far beyond
f.o.m., taking full advantage of the experience and lessons learned from
f.o.m., has been made.

And of course, what's really new is for me (smile) to engage in such an
attempt instead of just endlessly talking about how vitally important it is
to do it (smile). 

So I have been thinking a little bit about elementary physics and elementary
probability/statistics. What lessons from f.o.m. can be applied to the
foundations of these subjects?

Foundational clarification of even elementary quantum mechanics has of
course been the main topic for philosophers and others, as it appears to be
an entirely nonsensical theory with one particularly interesting attribute -
in its considerable realm, it predicts reality spectacularly well. This is
not a bad attribute to have (smile).

My own attitude towards this is that

1) major clarifying foundations for even elementary quantum mechanics is
probably hopeless until we get a far better grip on the foundations of less
mysterious physics;

2) in fact, we don't even have a good enough idea as to what foundations of
physical science should or could look like, in the first place.

As I will discuss in Foundational Thinking 2, one major role of foundational
thinking is to increase the quality of our knowledge. E.g., the creation of
formalisms such as propositional calculus, predicate calculus, Peano
Arithmetic, ZFC, etc., increases the quality of our mathematical knowledge
generally. The fact that most professional mathematicians have so much raw
mathematical talent - i.e., special affinity for mathematics - that they
feel that mathematical knowledge as is, without such formalisms, is already
at such a high level of quality that (they think that) these formalisms do
nothing for them, is completely besides the point.

On the other hand, Godel's main contributions lie in spectacular findings of
a new kind, launching off of these fundamental constructions. The story of
f.o.m. is very satisfying and dramatic - fundamental constructions made for
the purpose of increasing the quality of mathematical knowledge generally,
led to spectacular findings of a new kind, continuing, more or less, into
the present.

It seems obvious to me that outside of f.o.m., foundations is everywhere at
a pre-Godelian stage. In fact, it seems to be at a pre Fregean stage, and
maybe much earlier than that. So insisting that there be a Godelian payoff
for foundations outside mathematics, is absurdly premature.
Counterproductively premature. Imagine insisting on a Godelian payoff for
f.o.m. in the 19th century, and the effect that would have if adhered to!

So, where to begin? Let's try physics. We want to start by slowly crawling.

Special relativistic kinematics has some advantages for a slow crawl.
Undoubtedly, there are other good places for a slow crawl. They also demand
explicitly systematic foundational inquiry.

Special relativistic kinematics is the initial segment of special relativity
involving only space, time, and light transmission (only as it directly
impacts space, time). No particles, bodies, mass, energy, force, motion,
velocity, etc.  

The choice of special relativistic kinematics has these attributes, making
it a good place to start crawling:

1) it is almost universally considered to be fully understood, dead and
buried, with nothing new to say;

2) on the other hand, there is a substantial literature on it, even to this
day, mostly written by physicists in the name of physics education.

It may sound like 1) and 2) is the kiss of death for foundational work in
special relativistic kinematics - and of course this is absolutely true in
academia. I would not like to come up for tenure anywhere in any kind of
Department - mathematics, physics, or philosophy - based on any work, no
matter what the quality, in foundations of special relativistic kinematics!!

The advantage of 1) is that this is going to be a matter of pure quality of
knowledge, uncluttered by any nontrivial science. The physics, and the
related mathematics, are now regarded as utterly trivial (not in 1905).

The advantage of 2) is that, despite the triviality of relativistic
kinematics, there is obviously some reasonably widespread dissatisfaction
with the way it is taught in University courses. Hence we see all of these
publications, often citing classroom experiences, many of which are in the
American Journal of Physics, a journal explicitly devoted to physics
education. 

What is the stated dissatisfaction? My reading of it stems from the
antifoundational act of prematurely introducing the Minkowski inner product
(or even the Minkowski norm) on R x R^3, or even on R x R, and/or the
antifoundational act of prematurely introducing the Lorentz group of
transformations or the Poincare group of transformations from R x R^3 into R
x R^3, or even from R x R into R x R.

So most of these papers have titles like "Derivation of the Lorentz
transformations". There are perhaps - just a guess - several hundred
published papers with this theme starting soon after Einstein's 1905 paper,
continuing till the present. As I said earlier, the recent ones mostly give
the reason for publication as "educational".

OBSERVATIONAL AND THEORETICAL FOUNDATIONS.

I started off thinking about special relativistic kinematics (SRK) as a
target for what I call observational foundations. In fact, my intention
still is to give observational foundations for everything in physical
science. 

Generally, in observational foundations, one has only observers and
observations. Ideally, there are no theoretical notions such as event,
space, time, bodies, forces, motion, energy, etcetera. One builds such
notions by logically piecing together patterns in observations.

In observational foundations, the content of the theory one is giving
foundations for is given by what I call

*set of all possible world histories.*

Generally speaking, a world history is a well defined mathematical entity
which declares a (generally finite) number of distinct observers, and the
entire history of all observations made by those observers.

In standard philosophical terms, this is the

*set of all possible worlds (with only interpretations of observational
primitives) compatible with the underlying theory about which one is giving
observational foundations.*

In addition to giving mathematically perspicuous characterizations of the
possible world histories, one also wishes to give a set of simple powerful
axioms that must be satisfied by the world histories, so that the "models"
of these axioms are exactly the possible world histories (i.e., the ones
compatible with the underlying theory).

This should result in an important family of new observational formalisms.
In fact, formal systems in the sense of f.o.m.

But one still needs a clear, clean, workable, unambiguous formulation of the
underlying theory, with its natural theoretical concepts that are NOT
observational (it may or may not include the observational notions), in
order to prove that one has the correct determination of the set of all
possible world histories (i.e., those compatible with the underlying
theory). 

So the mathematical results will take the form

*a world history is compatible with this underlying theory if and only if it
has thus and such mathematical properties.*

The underlying theory should also be given a complete - in various senses -
axiomatization. Thus we have two axiomatizations - the observational
axiomatization and the theoretical axiomatization.

Corresponding to the theorem that thus and such world histories are exactly
those compatible with thus and such theory, we also should have detailed
relationships between the corresponding axiomatic theories that

*take on the nature of what we know intimately from f.o.m. as conservative
extension results and interpretability results. In particular, the
theoretical formalisms should be conservative extensions of the
observational formalisms.*

I already see how some principal preoccupations in f.o.m. such as
conservative extension and interpretability can fit into foundations of
physical science. However, they have to be adapted in certain respects that
would not be of any particularly high priority in f.o.m. E.g., there are
modifications that need to be made to handle the existence of physical
constants like c (speed of light in vacuum), which are best handled by the
introduction of a constant symbol c for an unknown positive real number.
This forces one to be extra careful with notions like completeness,
conservative extension, interpretability, and the like. Undoubtedly, these
adaptations will suggest some new theorems in f.o.m.

Thus we will see f.o.m. feeding into foundations of physical science, and
foundations of physical science feeding back into f.o.m.

In particular, when thinking about the required theoretical axiomatization
of special relativistic kinematics to be used in conjunction with the
to-be-developed observational axiomatization(s) of SRK, I saw the clear need
for a maximal principle.

I realized that this same kind of maximal principle was already employed by
Hilbert in his axiomatization of Euclidean geometry, but not in a first
order context. 

So I looked for an appropriate formulation of T(max) for first order
theories that would yield the complete first order axiomatizations of
Euclidean geometry and of real closed fields.

And if I need T(max), then why not formulate T(min)?

T(min) did not seem to be any kind of alternative way of generating complete
first order axiomatizations of Euclidean geometry and real closed fields and
the like. So what good is T(min)?

By a principle of duality (smile), if T(max) is so good, then T(min) must
also be good. 

So I guessed that T(min) was good for discrete math - Peano Arithmetic.

This is documented in posting #205.

And now we see this on the FOM email list:

On 1/21/04 2:04 PM, "Martin Davis" <martin at eipye.com> wrote:

> From Vladimir Lifschitz:
> 
> <<Right. T(min) is the early form of circumscription defined by John
> [McCarthy] in
> "Epistemological problems of Artificial Intelligence," Proc. IJCAI-77.
> Later he called it "domain circumscription." T(max) seems to be new.>>

By the way, did the T(min) people ever notice that it generates Peano
Arithmetic?

MATHEMATICS SHOULD NOT BE SEEN.

Notice that I did not say "mathematics should not be used."

By far the main reason why the physical science created/discovered one or
even many more centuries ago has always been essentially incomprehensible to
the vast majority of even highly intelligent people is the use of
substantial mathematical concepts in their presentation. The difficulty is
compounded by the liberal use of symbols, which happen to be generally
aesthetically disgusting. In fact, much of the relevant classical
mathematical notation is not only repulsive, but also logically incompetent.
This is understandable, given the context in which these
creations/discoveries where made.

Of course, modern mathematical treatments often employ better notation, but
at a cost. Another layer of mathematical complexity is added, and facility
with this extra layer of mathematical complexity is assumed - often
mathematicians are writing for other mathematicians.

Furthermore, this additional mathematical complexity does not have a rather
badly needed foundational exposition. See forthcoming "Foundational Thinking
2" for a discussion of foundational exposition. So this additional
mathematical complexity, which is in many ways a great improvement over any
classical mathematical treatment of physical science, is entirely
inaccessible to the vast majority of even highly intelligent people - and
looks to remain so for the indefinite future.

So the net effect is that even the century or older physical science is
doomed to remain essentially incomprehensible for the vast majority of even
highly intelligent people for the indefinite future.

However, this should all radically change, and in fact *will* radically
change.

There is no doubt that many people are aware of this difficulty and some try
to do something about it. One is Mermin, Cornell Physics Dept, with his
beautiful Physics For Poets.

These efforts are not radical enough, and don't go far enough. Part of what
needs to be done, in addition to these efforts, is a foundational exposition
of mathematics. (Again see "Foundational Thinking 2").

Generally speaking, every crucial mathematical concept serves some crucial
intellectual purpose and can be redefined as the unique mathematical concept
that serves that crucial purpose.

This allows us to *weave in* that crucial purpose in as part of the
axiomatization of physical science.

The crudest implementation of this idea is to simply take all of the
relevant crucial mathematical concepts as primitives, along with the
physical primitives, and thereby have a completely transparent
axiomatization.

This brute force implementation of the idea will be too unwieldy, and
therefore one must find key simplifications. Such key simplifications may
require a great deal of imaginative research and mathematical expertise.

Nevertheless, the *presentation* of the physical science thereby becomes
incomparably more transparent - after the simplification and synthesis
process takes hold.

In fact, in a sense, virtually NO mathematical complexity should be left in
the presentation. 

But where did the mathematical complexity go? Did it just simply vanish?

No, it is in the back room. Given certain minimally looking friendly
transparent axioms, how do we know that they have any strong consequences?
How do we know that they are "complete" in any appropriate sense? Those
proofs are in the Appendices.

One specific key idea that affects every aspect of this plan is the
uncovering of a unifying principle of "passing to the limit".

We are all familiar with the idea that when introducing integration of, say,
continuous real valued functions on closed intervals, we really only need to
know how to integrate the single function

f(x) = 1

on the closed unit interval [0,1]. This integrates to 1, and most
intelligent people can understand that (smile). In the back room, we apply
additivity, monotonicity, and a limit procedure, and then we have all that
we want. 

Similarly, we should only need to state the foundations of physical theories
in terms of what it says

*in the most trivial cases possible, where at least something nontrivial is
going on*

and then let the math take over in a uniform way, in the back room.

In this manner, the presentation of physical science should only display
(what looks to be) trivialities. At least, that is the goal.

Of course, this plan has not even been carried out explicitly systematically
for mathematics itself, although there are plenty of developments in this
direction lying around. A systematic realization of this is an integral part
of the foundational exposition of mathematics.

The next numbered posting will focus on specifics regarding SRK.

*********************************************

I use http://www.mathpreprints.com/math/Preprint/show/ for manuscripts with
proofs. Type Harvey Friedman in the window.
This is the 206th in a series of self contained numbered postings to
FOM covering a wide range of topics in f.o.m. The list of previous
numbered postings #1-149 can be found at
http://www.cs.nyu.edu/pipermail/fom/2003-May/006563.html  in the FOM
archives, 5/8/03 8:46AM. Previous ones counting from #150 are:

150:Finite obstruction/statistics  8:55AM  6/1/02
151:Finite forms by bounding  4:35AM  6/5/02
152:sin  10:35PM  6/8/02
153:Large cardinals as general algebra  1:21PM  6/17/02
154:Orderings on theories  5:28AM  6/25/02
155:A way out  8/13/02  6:56PM
156:Societies  8/13/02  6:56PM
157:Finite Societies  8/13/02  6:56PM
158:Sentential Reflection  3/31/03  12:17AM
159.Elemental Sentential Reflection  3/31/03  12:17AM
160.Similar Subclasses  3/31/03  12:17AM
161:Restrictions and Extensions  3/31/03  12:18AM
162:Two Quantifier Blocks  3/31/03  12:28PM
163:Ouch!  4/20/03  3:08AM
164:Foundations with (almost) no axioms, 4/22/0  5:31PM
165:Incompleteness Reformulated  4/29/03  1:42PM
166:Clean Godel Incompleteness  5/6/03  11:06AM
167:Incompleteness Reformulated/More  5/6/03  11:57AM
168:Incompleteness Reformulated/Again 5/8/03  12:30PM
169:New PA Independence  5:11PM  8:35PM
170:New Borel Independence  5/18/03  11:53PM
171:Coordinate Free Borel Statements  5/22/03  2:27PM
172:Ordered Fields/Countable DST/PD/Large Cardinals  5/34/03  1:55AM
173:Borel/DST/PD  5/25/03  2:11AM
174:Directly Honest Second Incompleteness  6/3/03  1:39PM
175:Maximal Principle/Hilbert's Program  6/8/03  11:59PM
176:Count Arithmetic  6/10/03  8:54AM
177:Strict Reverse Mathematics 1  6/10/03  8:27PM
178:Diophantine Shift Sequences  6/14/03  6:34PM
179:Polynomial Shift Sequences/Correction  6/15/03  2:24PM
180:Provable Functions of PA  6/16/03  12:42AM
181:Strict Reverse Mathematics 2:06/19/03  2:06AM
182:Ideas in Proof Checking 1  6/21/03 10:50PM
183:Ideas in Proof Checking 2  6/22/03  5:48PM
184:Ideas in Proof Checking 3  6/23/03  5:58PM
185:Ideas in Proof Checking 4  6/25/03  3:25AM
186:Grand Unification 1  7/2/03  10:39AM
187:Grand Unification 2 - saving human lives 7/2/03 10:39AM
188:Applications of Hilbert's 10-th 7/6/03  4:43AM
189:Some Model theoretic Pi-0-1 statements  9/25/03  11:04AM
190:Diagrammatic BRT 10/6/03  8:36PM
191:Boolean Roots 10/7/03  11:03 AM
192:Order Invariant Statement 10/27/03 10:05AM
193:Piecewise Linear Statement  11/2/03  4:42PM
194:PL Statement/clarification  11/2/03  8:10PM
195:The axiom of choice  11/3/03  1:11PM
196:Quantifier complexity in set theory  11/6/03  3:18AM
197:PL and primes 11/12/03  7:46AM
198:Strong Thematic Propositions 12/18/03 10:54AM
199:Radical Polynomial Behavior Theorems
200:Advances in Sentential Reflection 12/22/03 11:17PM
201:Algebraic Treatment of First Order Notions 1/11/04 11:26PM
202:Proof(?) of Church's Thesis 1/12/04 2:41PM
203:Proof(?) of Church's Thesis - Restatement 1/13/04 12:23AM
204:Finite Extrapolation 1/18/04 8:18AM
205:First Order Extremal Clauses 1/18/04 2:25PM

Harvey Friedman