Identicality and discreteness

CSGnet Archive CSG1996
1

Hans Blom wrote:

Even more basic is the notion "identical" itself. Although ancient
wisdom says that you cannot step into the same river twice, yet we
have the urge to see similarities and invent classes or categories.
These are utterly subjective: what may be the same for one person
may be extremely different for someone else. One might say that this
has to do with the perceptual input functions that we have developed.
In another sense, we are frequently helped by discovering
commonalities (going up a level of abstraction, one could say).

...

In one sense, no experiment can ever be reproduced. In another sense,
we talk about "repeating" all the time. And that is useful; only
where we assume invariance can we perform some sort of "averaging"
and obtain "knowledge" that can provide some sort of prediction of
how "identical" operations will have "identical" results. Thus,
classifications are necessarily approximations or perspectives. But
we cannot do without them.

The whole reason why we have developed discrete, symbol-based notational
systems is to be able to say, unequivocally and unambiguously, that
2 outcomes are "the same" or "different". If the means for detecting
and distinguishing the various discrete outcomes are reliable enough,
such that all observers always agree on the result (e.g. that a
"2" is showing on the computer screen rather than a "6"),
then experiments CAN BE completely replicable
(depending on the material processes underlying the
outcomes of measurements). This also applies in the realm of
(finite, physically-realized) formal operations,
where we can always, unequivocally and unambiguously, reliably agree
what "2+2" means and that the outcome of this operation is "always"
(within humanly relevant time frames) equivalent to "4"
(and not "5" or "9"). What allows this reliability
is the construction and mutual calibration of sensors and symbols
such that the sensors of all observers register the same outcome
whenever a given symbol is presented for inspection.

We need not have any knowledge whatsoever of the processes underlying
the action of the sensors in order to be able to bring them into this
kind of correspondence (as long as we have ways of steering the
sensors, of tuning them), and this is what is truly powerful about our
ability as a species to construct the elaborate symbol-systems that
we do. The sensors themselves implement the perceptual categories and
the categories of different observers are brought into registration
("calibration") by tuning of sensors.

Everything need not be mired in vagary and ambiguity; there
are (constructed) realms in which things can be (made) crystal clear.

I have said nothing here about incomplete models or stochastic
processes or what to do in those situations, only that the
apparently indeterminate nature of the outcomes is not necessarily
due to inherent uncertainties or "subjectivities" or "noise" in the
observers.

Peter Cariani

2

[Hans Blom, 960709c]

(Peter Cariani <peter@EPL.MEEI.HARVARD.EDU>)

The whole reason why we have developed discrete, symbol-based notational
systems is to be able to say, unequivocally and unambiguously, that
2 outcomes are "the same" or "different". If the means for detecting
and distinguishing the various discrete outcomes are reliable enough,
such that all observers always agree on the result (e.g. that a
"2" is showing on the computer screen rather than a "6"),
then experiments CAN BE completely replicable
(depending on the material processes underlying the
outcomes of measurements).

Sure, if we have exactly the same understanding of a problem and if
the problem's solution is unique, we will, if we also know the ways
in which to solve the problems of the domain, probably arrive at the
same solution. That is beautiful but, regrettably, possible only in
very restricted domains in which all "real world" connotations have
been carefully removed from the symbols and operations that are used.
As Einstein (quoted by Bruce Gregory) said: mathematics (or symbolic
logic) has little to do with reality (whatever that may be).

I was talking about how to proceed from a bare perception to a
symbolic "2" or "6". It is possible, for instance, to construct a
symbol that, without hesitation, will be given two different,
mutually exclusive meanings. Try something like a hand-written

      5
    a b c
      7

in which the middle symbol is a bit fuzzy. You can do that in a way
in which the symbol is "recognized" without hesitation as both a "b"
and a "6", depending upon whether the row or the column is read. The
"meaning" of a symbol thus seems to greatly expect on its context,
i.e. on our expectations.

Everything need not be mired in vagary and ambiguity; there
are (constructed) realms in which things can be (made) crystal clear.

Yes, I agree. In one sense, this is true. Yet, when you read about the
philosophophy of mathematics, the "crystal clear" concepts of mathe-
matics are built on pretty fuzzy foundations. What is a point? What
is a number? In his fat two-volume "Principia Mathematica", Bertrand
Russell needs all of the first volume to define the number 1. How
"crystal clear" can such a definition be?

Paradox?

Greetings,

Hans Blom

3

(Peter Cariani July 9 1996)

[Hans Blom, 960709c]
(Peter Cariani <peter@EPL.MEEI.HARVARD.EDU>)

> The whole reason why we have developed discrete, symbol-based notational
> systems is to be able to say, unequivocally and unambiguously, that
> 2 outcomes are "the same" or "different". If the means for detecting
> and distinguishing the various discrete outcomes are reliable enough,
> such that all observers always agree on the result (e.g. that a
> "2" is showing on the computer screen rather than a "6"),
> then experiments CAN BE completely replicable
> (depending on the material processes underlying the
> outcomes of measurements).

Sure, if we have exactly the same understanding of a problem and if
the problem's solution is unique, we will, if we also know the ways
in which to solve the problems of the domain, probably arrive at the
same solution. That is beautiful but, regrettably, possible only in
very restricted domains in which all "real world" connotations have
been carefully removed from the symbols and operations that are used.
As Einstein (quoted by Bruce Gregory) said: mathematics (or symbolic
logic) has little to do with reality (whatever that may be).

I couldn't agree more. But when we do want a reliable toy world,
we know how we can built one. This is all that I'm saying

I was talking about how to proceed from a bare perception to a
symbolic "2" or "6". It is possible, for instance, to construct a
symbol that, without hesitation, will be given two different,
mutually exclusive meanings. Try something like a hand-written

      5
    a b c
      7

in which the middle symbol is a bit fuzzy. You can do that in a way
in which the symbol is "recognized" without hesitation as both a "b"
and a "6", depending upon whether the row or the column is read. The
"meaning" of a symbol thus seems to greatly expect on its context,
i.e. on our expectations.

It CAN depend upon our expectations, and much is made of this in
cognitive science. But as often as not, we do perceive things that
we have not a priori "expected". Through effectors we influence the
world so as to control our perceptions, but this is different from
controlling our sensors themselves such that we and only we
determine their output readings. If we completely control our own
sensors, then there is no longer any linkage with the world,
and we no longer can influence the world outside ourselves in
any coherent way.

> Everything need not be mired in vagary and ambiguity; there
> are (constructed) realms in which things can be (made) crystal clear.

Yes, I agree. In one sense, this is true. Yet, when you read about the
philosophophy of mathematics, the "crystal clear" concepts of mathe-
matics are built on pretty fuzzy foundations. What is a point? What
is a number? In his fat two-volume "Principia Mathematica", Bertrand
Russell needs all of the first volume to define the number 1. How
"crystal clear" can such a definition be?

Paradox?

Well, I agree that the foundations of mathematics is currently on
pretty shaky ground, but I believe this is because Russell and
Hilbert admitted potential infinities (i.e. "ill-defined"
entities) into their systems. If you
limit your notational systems to finite string lengths,
then everything is a finite state automaton, everything is
surveyable, and consistency can be established. No fuzziness
there. It's when platonic entities are introduced that the
foundations begin to crumble. We can have "perfectly" consistent
notation-systems if we want them; they just have to be
finite ones. We can now build digital computers that
will not make an error in our lifetimes.

What would a PCT account of the foundations of
mathematics look like? I think it would look alot like
a PCT account of language, in the sense that
groups of (limited) observers come to agreements
about rules for manipulating symbols through
ongoing, co-evolutionary behavioral interactions.
We would start with finite observers capable of making
finite numbers of distinctions on tokens (or on analog
signals if one prefers them) and capable of reliably
performing operations on those tokens so as to implement "rules".
We can probably all imagine control systems that can "control"
for the perception of a reliable sequence of perceived states
(a perceptual sequence). It would take a while, but PCT
could probably eventually be linked up with
constructivist (finitist) mathematics. What good this
would do us all, though, is an open question.

Peter

4

[Hans Blom, 960710b]

(Peter Cariani July 9 1996)

I couldn't agree more. But when we do want a reliable toy world,
we know how we can built one. This is all that I'm saying

Agreed. Our self-built toys are firmly in our grip, as long as we
keep things simple. But don't build something too complex: it may
show quite unexpected behavior. Frankenstein-like stories keep
haunting humanity...

It CAN depend upon our expectations, and much is made of this in
cognitive science. But as often as not, we do perceive things that
we have not a priori "expected". Through effectors we influence the
world so as to control our perceptions, but this is different from
controlling our sensors themselves such that we and only we
determine their output readings. If we completely control our own
sensors, then there is no longer any linkage with the world,
and we no longer can influence the world outside ourselves in
any coherent way.

Fully agreed. This is profound. I have often wondered what would
happen if we suddenly encountered something that was fully
unexpected, something that we could never have anticipated. I do not
mean the sensor part: if our sensors lack the capability to detect
it, it simply wouldn't be there. But what would happen when, say, an
alien from another planet suddenly stood in your living room? If it
looked like a dog, you would treat it like a dog. If it looked like a
little green man, a spider, an ape, we would react according to our
pre-existing knowledge and our experience that things that look
similar must BE similar. How would a dog be able to tell us (s)he is
not a dog? Regrettably, I cannot imagine the fully unexpected.

Well, I agree that the foundations of mathematics is currently on
pretty shaky ground, but I believe this is because Russell and
Hilbert admitted potential infinities (i.e. "ill-defined"
entities) into their systems.

That is a different problem, maybe. In Principia Mathematica's first
part, the "ill-defined" entity is not infinity but zero. Once you
have defined the number zero (how would you do that? from the "empty
set"?), the number one is easy once you have introduced ordering and
the concept of successor. The basic problem is, I think, to base the
rigid notions of mathematics on the fluid notions of our intuitions.

If you
limit your notational systems to finite string lengths,
then everything is a finite state automaton, everything is
surveyable, and consistency can be established.

Yes, I think that the introduction of "algorithms" has helped a lot
to make things clearer. But even before that, some mathematicians
(e.g. Brouwer) insisted that proofs should only be considered valid
if a CONSTRUCTION is possible. Strangely, maybe, his math is called
"intuistionistic". It is also less powerful, because it can prove
less. This includes some results that most mathematicians consider
trivially true and that are easily proved by e.g. using the method of
the "excluded middle".

We can now build digital computers that will not make an error in
our lifetimes.

But computers, too, have the same limitation that they cannot compute
or "reason" when infinities are involved. That may be a good thing.
Maybe we should abolish those parts of math that are not computable.
On second thought, wouldn't that eliminate Euclidean geometry as
well?

Greetings,

Hans