The Return of the Empirical Subject in the Philosophy of Mathematics - Bartosz Brożek

The lecture of Bartosz Brożek entitled 'The Return of the Empirical Subject in the Philosophy of Mathematics' delivered during the 'Philosophy in Science' Copernicus Center International Seminar.The goal of the seminar was to analyze philosophical issues in science from both historical and analytical perspectives.

Philosophy in Science is a method of philosophical reflection which is carried out in the context of the sciences. Philosophical ideas have often influenced -- and continue to influence -- the development and evolution of scientific theories; traditional philosophical problems are intertwined with empirical theories and the assumptions of science should become the subject of fruitful debate. Philosophy in Science should be distinguished from philosophy of science: while the latter views science 'from beyond', the former stems from scientific practice. The conception of philosophy in science was developed by Michael Heller, who clarified its basic principles in a paper published in 1986 entitled "How is Philosophy in Science possible?"

The seminar is funded from the research grant The Limits of Scientific Explanation, awarded by The John Templeton Foundation.

[DarkMoonDroid]

3...
In most cases those lines were connected in a sort of cursive to create more complex squiggles. One line is 1. Two lines are connected at the ends with a curved line and you get a 2. Three lines are connected in the same way and you get a 3. I'm sure you've seen that TED talk with Jill Bolte Taylor who had a stroke and when she went to make a phone call, she had lost the memory of the meanings of the squiggles, but she could look at them and see the similarity of form.

[DarkMoonDroid]

When my kids were 3 and 6, I was teaching them math using a system that I invented. By the time my daughter was ready to go to kindergarten, she already understood addition, subtraction, multiplication and division and what a prime number was. I used some weird system of circles and dots. I forget how I did it, but she understood it. It was not memorization. She was too young for that. She understood what was actually happening.
How did that happen?

[DarkMoonDroid]

The idea that language is what moves the cognition past the 4-barrier seems good. I'm not sure it's language itself, but the even more fundamental processes and structures upon which language depends. It is available to see for anyone who chooses to look that language evolved from images. In the West, the door to the tent was stylized further and further until it became a "D". In the East, there are similar things happening about which I don't have anything I can quote handy. :-/

[DarkMoonDroid]

2...
In most ancient written languages, numbers were slightly different. One was one vertical or horizontal line. Two was two lines. Three was three lines, etc. What is common in both cases was that the image carried a meaning. Every time you walk a sheep thru the gate, you draw a line. Or every day that passes between the first day a woman starts bleeding to the next day she starts bleeding, she carves a line on a bone.

[fergoesdayton]

John Von Neumann, was critical of abstract philosophy of mathematics, because abstractions are grounded in our empirically novel selection of items. Von Neumann viewed this freedom in selection to be a form of art. Every time the rationalist mind may ask, what comes next in the set: 2, 12, 211, 1112, an empiricist can reply that they brought the numbers together in the first place.

[DarkMoonDroid]

4...
She had lost the ability to hold abstractions, but she could copy an image or a squiggle. She didn't have access to "threeness". She had access to 3-shaped squiggles.
This suggests links with both language & physical, concrete objects.
I was also thinking about why 4? Is 4 actually fourness or is it two sets of two. Much like cell-division, a dyad is easy to understand. Duplication is at the foundation of all biological growth.

[fergoesdayton]

'Abstraction' in this sense that it is useful in math, is usually rooted in empirical thinking. That is to say, those who believe in circles, owe it to the moon.

[DarkMoonDroid]

I have noticed that the people who love math who I've actually talked to do not seem to respect the role of the Subject in their field. They seem to believe that the Subject is excluded from Mathematics because 2 + 2 = 4 whether you're a computer or a person. It's factualness seems to instill some sense of freedom from their messy or confused self-sense. Er whatever...
But when you put up the page at 01:28 with the 3 types of Subjects, it all became really obvious to me what has been missing.

[DarkMoonDroid]

Incidentally, I suspect that I am of the "transcendental subject" type. I just never went to school and, ironically, I don't have the "language".
:-/

[DarkMoonDroid]

5...
Perhaps the child can REALLY only count up to 3, but 4 is 2 dyads. Mother and child is the first dyad - and that is implicit because it takes quite a while to understand that both are separate selves and they are not sharing the same consciousness. But the second dyad is easier to see: Mom and Dad. Provided you have a Mom and Dad... These are what carry meaning in a child's world for a long time. Esp. girls. Boys learn about non-conscious objects sooner. But the dyad is very fundamental.

[bristol]

The three levels of reality in Platonic Physics
Roger B Clough, National Institute of Standards and Technology (Retired)
(11-28-2014)

Abstract
Here we combine the top-down metaphysics of Plato and Leibniz with the inside-out categories of C S Peirce to enable us to view the world in a new, more useful light, simultaneously from two perspectives, and in more detail than Leibniz's pre-established harmony.  The top down structuring from  Plato and Leibniz allows us to view the world as it is: governed cybernetically by thought from the top singularity (the One, comparable to a computer processing unit), rather than from the ungoverned perspective of current science.  This allows us not only to understand the world properly, but  to structure the world cybernetically. with all creation, perception and control coming in the form of thought from the top down, but inside out using C S Peirce's three categories. 

1. Introduction. While C S Peirce is well known to the philosophy of science, the worlds of Plato and his follower Leibniz have been less explored for such purposes.  Plato was an Idealisti and Arthur Eddington spent much of his life adapting Plato to science, but his use of Mind  in a world thoroughly established in materialism ihas largely blocked exploration of the use of Mind cybernetically, as a singular, mental control point, so that the current world of science is only governed, if at all, in fiefdoms. But more significantly, materialism and a lack of a single cybernetic control from top down has hindered the develepment of an understanding to consciousness, thought and the role and nature of the self.  For example, Dennett in his explanation of consciences does not have a perceiver (or at best a fancifal and abstract invention of one). Moreover the perceiver, to obviate the homunculus with homunculus problem, must be on a higher ontological level, and which has to be a living singular entity, not an abstract reference. By application of Leibniz and Plato and common sense as well, , we see that the perceiver must be singular-- the One, the cybernetic Perceiver and control point, the central processing unit, to use a computer analogy.
The learning curve on Plato-Leibniz is a bit steep at first, foreign to most physical scientists because of their unfamiliar top down control, which is also done indirectly by thought rather than directly by physical interaction, but also because of  Leibniz's unfamiliar  spreadsheet  style ontology, using not atoms but complete concepts called monads, which can be nested like sets. That would seem to render Leibniz more understandable to mathematicians and computer science, but his thinking in terms of substances and monads can be off-putting.  Once these are understood (through his Monadology [ ]) and if one sticks to the elementary particles scale (the particles are both substance and monads) one can proceed fairly smoothly.

2. The three levels

FIRSTNESS -FIRST PERSON (I) -Mind- The One, the Monarch- this is the realm of Plato's Mind. It is life itself, pure nonphysical intelligence. Purely subjective, timeless and spaceless -     with innate knowledge and a priori memory, containing the pre-established harmony, necessary logic, numbers - the womb of the WHAT.   Mind creates all, perceives all, controls all. Thus the individual mind controls the brain,    not the reverse. Mind plays the brain like a violin.

SECONDNESS – SECOND PERSON (YOU RIGHT HERE) ental objects so both subjective +objective- The Many.  In this, the WHAT separates  from  Mind and becomes a  HERE.  Accordingly. Heidegger referred to existence as "dasein". "Being here."
    According to Leibniz, all monads are alive to various degrees.   There are of three gradations of life in these,     according to Leibniz:

    a) Bare, naked monads, which we can think of as purely physical  ( Eg, a fundamental particle).

    b) Animal and vegetative monads, which Leibniz calls souls, which can have feelings, but little intellect.

   c) Spirits (corresponding to humans), which have, in addition, intellectual capacities.  Mind transforms physical signals in nerves and neurons into experiences. If Mind then reperceives or reflects on these experiences, they are said to be thoughgt or apperceived. To be apperceived is to be made conscious. Thus consciousness is the product of thought. Intentions are also made in the same way, so that we caqn say that thoughts are intentions by Mind.
The human brain is a monad which contains as subsets,   mental capacities.  Neuroscience tells us that there is binding between  monads for parts and functions of the brain, but since monads cannot act directly on each other, this binding must be indirect, through the sequential updates of the perceptions and appetites of the subfunction monads. These must be made by Mind, either directly or through the preestablished  harmony PEH). Unfortunately the Stanford Leibniz site on Leibniz makes no mention of the action of  Mind on the individual mind,   IMHO a gross shortcoming.

Sensory signals and signals for feelings must also go through such a binding process.  In a sense, the   binding process plays the role of a self, but in conventional neuroscience self is a function of  the brain, rather than the other way round, as common sense suggests and the intentionality of  self or mind  proves, along with the need for a PEH.

This shortcoming in conventional understanding of the brain becomes all the more nagging if we  consider thinking, which is closely related to apperception, because it must be conscious.Thinking, we submit, consists of consciously manipulating and comparing such  apperceptions.  

 Through Mind, with its potentially infinite wisdom and intelligence,   intuitions and thoughts can arise spontaneously in the individual mind.  If these are to be immediate and/or original,   it is reasonable to believe that they originate in Mind, rather than  indirectly through separate although bound parts         of the brain. Anyone who has experienced a vocal duet in which the vibratos are in phase should become         convinced of this.
       
        Mind is the monarch of the intelligent mind, which controls the brain. Mind plays the brain like a violin.        Mind is also is able to focus on a thought for a brief period,   within the context of one's memory and universal memory,         for purposes of thinking an comparison, making the biological brain and its  complex bindings seem hopelessly         indirect and subject to confusion.


THIRDNESS – THIRD PERSON (IT OVER THERE) Corresponding physical objects as is appropriate- -here the object is born or emittted     from the monad--and emerges into spacetime as a particle, becoming completely objective,     a WHAT+ HERE +WHEN.,   In addition the Thirdness of a private thought or experience is its     public expression in some appropriate form.

3. Conclusions
This format allows us to examine quantum phenomena from inside out and perception, thinking and consciousness ontologically- from physical nerve signals to mental experiences such as thought, consciousness, and cognition.  It also avoids problem encountered in “bottom-up” science, such as complexity and emergence, if for no other reason than there is no apparent way of conceiving of a singular control point at the bottom.

--
Dr. Roger B Clough NIST (retired, 2000).

[bristol]

Mathematics as a level of reality (Secondness) in platonic physics

The three levels of reality in platonic physics are given below.
Mathematics, like ideas in general -- that is to say, mental objects-- is a form of Secondness,
which in turn is an object of Mind or Firstness.   Mind is the timeless, spaceless central
domain of Plato's metaphysics, containing a priori certainties such as necessary logic,
arithmetic and numbers, as well as Leibniz's pre-established harmony. It does all of the
thinking, controlling and perceiving of the universe. Nothing is done without it
and its control is cybernetic.

Mathematics, as Secondness,   is simply a mental object nlike monads are, but
it is not monadic in that it has no correponding physical bodies. Insterad, it is a
special language with a special grammar, in particular arithmetic and logic,   but on its
 own can do nothing, create nothing, or achieve anything anymore than a human
can using mathematics as a tool of thought. Or Shakespeare could with the english language.

This is in line, if we consider mathematics as a language, with the idea of computationalism


" In philosophy, a computational theory of mind names a view that the human mind or the human brain (or both)
is an information processing system and that thinking is a form of computing."

which unfortunately has been construed by some to give anthropomorphic or god-like powers to mathematics.
The universe was not constructed by mathematics alone, as it is mental, not physical.

While this passivity may seem to some to diminish the power of mathematics, it in fact frees it from
the limitations of the outer world, such as the limitations of Leibniz's pre-established harmony,
and limits it only to its inherent logic and arithmetical constraints. This is particularly useful in
abstract mathematics.

In addition, due to what has been referred to as "the unreasonable effectiveness" of mathematics
in the world, it appears that mathematics is predominantly the language of the pre-established harmony
of Leibniz.

In addition, all of Plato's philosophy of mathematics should hold.




The three levels of reality in platonic physics


FIRSTNESS -FIRST PERSON (I) -Mind- The One, the Monarch- this is the realm of Plato's Mind. It is life itself, pure nonphysical intelligence. Purely subjective, timeless and spaceless -     with innate knowledge and a priori memory, containing the pre-established harmony, necessary logic, numbers - the womb of the WHAT.   Mind creates all, perceives all, controls all. Thus the individual mind controls the brain,    not the reverse. Mind plays the brain like a violin.

SECONDNESS – SECOND PERSON (YOU RIGHT HERE) ental objects so both subjective +objective- The Many.  In this, the WHAT separates  from  Mind and becomes a  HERE.  Accordingly. Heidegger referred to existence as "dasein". "Being here." Some of these objects, such as ideas, or mathematics, are not monads, since they have no corresponding physical bodies.
    According to Leibniz, all monads are alive to various degrees.   There are of three gradations of life in these,     according to Leibniz:

    a) Bare, naked monads, which we can think of as purely physical  ( Eg, a fundamental particle).

    b) Animal and vegetative monads, which Leibniz calls souls, which can have feelings, but little intellect.

   c) Spirits (corresponding to humans), which have, in addition, intellectual capacities.  Mind transforms physical signals in nerves and neurons into experiences. If Mind then reperceives or reflects on these experiences, they are said to be thoughgt or apperceived. To be apperceived is to be made conscious. Thus consciousness is the product of thought. Intentions are also made in the same way, so that we caqn say that thoughts are intentions by Mind.
The human brain is a monad which contains as subsets,   mental capacities.  Neuroscience tells us that there is binding between  monads for parts and functions of the brain, but since monads cannot act directly on each other, this binding must be indirect, through the sequential updates of the perceptions and appetites of the subfunction monads. These must be made by Mind, either directly or through the preestablished  harmony PEH). Unfortunately the Stanford Leibniz site on Leibniz makes no mention of the action of  Mind on the individual mind,   IMHO a gross shortcoming.

Sensory signals and signals for feelings must also go through such a binding process.  In a sense, the   binding process plays the role of a self, but in conventional neuroscience self is a function of  the brain, rather than the other way round, as common sense suggests and the intentionality of  self or mind  proves, along with the need for a PEH.

This shortcoming in conventional understanding of the brain becomes all the more nagging if we  consider thinking, which is closely related to apperception, because it must be conscious.Thinking, we submit, consists of consciously manipulating and comparing such  apperceptions.  

 Through Mind, with its potentially infinite wisdom and intelligence,   intuitions and thoughts can arise spontaneously in the individual mind.  If these are to be immediate and/or original,   it is reasonable to believe that they originate in Mind, rather than  indirectly through separate although bound parts         of the brain. Anyone who has experienced a vocal duet in which the vibratos are in phase should become         convinced of this.
      
        Mind is the monarch of the intelligent mind, which controls the brain. Mind plays the brain like a violin.        Mind is also is able to focus on a thought for a brief period,   within the context of one's memory and universal memory,         for purposes of thinking an comparison, making the biological brain and its  complex bindings seem hopelessly         indirect and subject to confusion.


THIRDNESS – THIRD PERSON (IT OVER THERE) Corresponding physical objects as is appropriate- -here the object is born or emittted     from the monad--and emerges into spacetime as a particle, becoming completely objective,     a WHAT+ HERE +WHEN.,   In addition the Thirdness of a private thought or experience is its     public expression in some appropriate form.

3. Conclusions
This format allows us to examine quantum phenomena from inside out and perception, thinking and consciousness ontologically- from physical nerve signals to mental experiences such as thought, consciousness, and cognition.  It also avoids problem encountered in “bottom-up” science, such as complexity and emergence, if for no other reason than there is no apparent way of conceiving of a singular control point at the bottom.

--
Dr. Roger B Clough NIST (retired, 2000).

[bristol]

On a new birth of philosophy from the ashes of materialism

Materialism is the doctrine that nothing exists except matter and its movements and modifications.
This doctrine in its modern form seems to have originated from the secular philosophers of the
Enlightenment, and from the writings of Voltaire and others seems to have become established
in western thinking together with the secularization of society and its opposition to the power of the
Church. 

The critical turn of thinking appears to have been due to an incompleteness in the  metaphysics of
Descartes. Descartes, for all of his originality and brilliance, overlooked the integration of mind
and body, as noted by Leibniz, by dividing reality into two completely distinct realms, one
of extension (the body) and one of mind, which is non-extended.  This worked out well in practice,
by accomodating Newton's new mechanics, since his mechanics only apparently dealt with the physical world, and freed science from dealing with mind (and divinity) by simply ignoring it. Today,
with the advent of quantum mechanics, we know that this is not true, for
quanta are mental, not physical, since they are not independently in spacetime.

 In materialistic thinking, the mind is a product of the brain and  controlled by it.
This however cannot explain intentional acts, which originate in mind. It also
allows materialistic thinkers to ignore concepts such as the soul or divinity,
giving justification for secularism, and opening up the possibility of dialectical
materialism.

Leibniz pointed out that matter, since causality must be mental and not physical (since there are
no such physical entities as momentum, for example) must have some mental correspondent.
Leibniz called this mental correspondent the monad.  An example of a monad is a quantum.

Another serious problem with materialism is that physical entities in spacetime are contingent,
meaning that they are not permanent and fixed, as Bertrand Russell thought they were
in his theory of descriptions. They are thus poor, ephemeral referents,   since they both
move and continually change.

An example of a possible correction to materialism is given below, although obviously
others might be able to do better.


The three levels of reality in platonic physics
FIRSTNESS -FIRST PERSON (I) -Mind- The One, the Monarch- this is the realm of Plato's Mind. It is life itself, pure nonphysical intelligence. Purely subjective, timeless and spaceless -     with innate knowledge and a priori memory, containing the pre-established harmony, necessary logic, numbers - the womb of the WHAT.   Mind creates all, perceives all, controls all. Thus the individual mind controls the brain,    not the reverse. Mind plays the brain like a violin.

SECONDNESS – SECOND PERSON (YOU RIGHT HERE) ental objects so both subjective +objective- The Many.  In this, the WHAT separates  from  Mind and becomes a  HERE.  Accordingly. Heidegger referred to existence as "dasein". "Being here." Some of these objects, such as ideas, or mathematics, are not monads, since they have no corresponding physical bodies.
Our personal minds (small m) are made up of intentional objects (Secondnesses).

Similarly, Russell did not accept the importance of Mind and its filtering of
all knowledge, even public knowledge, into two forms: a) the
larger form, more public and less personally influenced,   of Mind,
and b) the local personal experiences of individual minds (small m),
which is knowledge by acquaintance.  Both of these forms are
ingerent in Leibniz's metaphysics.

Thus Russell's analytic philosophy only treats of a) the larger, public
knowledge of Mind, while all of our individual contents of mind (small m)
are type b), personal knowledge, experiential knowledge, knowledge by
acquaintance.

In contrast, both Brentano and Meinong, no doubt because they were
psychologists, were more interested in and studied b) rather than a).
The objects of b) are  objects of personal intention, as is all thought.
So Secondness might be said to consists of two two types of mental
objects, i)  nonmonadic intendeds and b) monads.
 




    According to Leibniz, all monads are alive to various degrees.   There are of three gradations of life in these,     according to Leibniz:

    a) Bare, naked monads, which we can think of as purely physical  ( Eg, a fundamental particle).

    b) Animal and vegetative monads, which Leibniz calls souls, which can have feelings, but little intellect.

   c) Spirits (corresponding to humans), which have, in addition, intellectual capacities.  Mind transforms physical signals in nerves and neurons into experiences. If Mind then reperceives or reflects on these experiences, they are said to be thoughgt or apperceived. To be apperceived is to be made conscious. Thus consciousness is the product of thought. Intentions are also made in the same way, so that we caqn say that thoughts are intentions by Mind.
The human brain is a monad which contains as subsets,   mental capacities.  Neuroscience tells us that there is binding between  monads for parts and functions of the brain, but since monads cannot act directly on each other, this binding must be indirect, through the sequential updates of the perceptions and appetites of the subfunction monads. These must be made by Mind, either directly or through the preestablished  harmony PEH). Unfortunately the Stanford Leibniz site on Leibniz makes no mention of the action of  Mind on the individual mind,   IMHO a gross shortcoming.

Sensory signals and signals for feelings must also go through such a binding process.  In a sense, the   binding process plays the role of a self, but in conventional neuroscience self is a function of  the brain, rather than the other way round, as common sense suggests and the intentionality of  self or mind  proves, along with the need for a PEH.

This shortcoming in conventional understanding of the brain becomes all the more nagging if we  consider thinking, which is closely related to apperception, because it must be conscious.Thinking, we submit, consists of consciously manipulating and comparing such  apperceptions.  

 Through Mind, with its potentially infinite wisdom and intelligence,   intuitions and thoughts can arise spontaneously in the individual mind.  If these are to be immediate and/or original,   it is reasonable to believe that they originate in Mind, rather than  indirectly through separate although bound parts         of the brain. Anyone who has experienced a vocal duet in which the vibratos are in phase should become         convinced of this.
       
        Mind is the monarch of the intelligent mind, which controls the brain. Mind plays the brain like a violin.        Mind is also is able to focus on a thought for a brief period,   within the context of one's memory and universal memory,         for purposes of thinking an comparison, making the biological brain and its  complex bindings seem hopelessly         indirect and subject to confusion.


THIRDNESS – THIRD PERSON (IT OVER THERE) Corresponding physical objects as is appropriate- -here the object is born or emittted     from the monad--and emerges into spacetime as a particle, becoming completely objective,     a WHAT+ HERE +WHEN.,   In addition the Thirdness of a private thought or experience is its     public expression in some appropriate form.

3. Conclusions
This format allows us to examine quantum phenomena from inside out and perception, thinking and consciousness ontologically- from physical nerve signals to mental experiences such as thought, consciousness, and cognition.  It also avoids problem encountered in “bottom-up” science, such as complexity and emergence, if for no other reason than there is no apparent way of conceiving of a singular control point at the bottom.
--
Dr. Roger B Clough NIST (retired, 2000).

[bristol]

A philosophy of mathematics in accord with Plato-Leibniz

Plato-Leibniz (PL) is an idealistic philosophy in which Plato's Mind or the One governs Leibniz Many.
Mathematicians, with some important exceptions, such as Godel, Frege and Leibniz, are usually bad philosophers.
To give a glaring example, Bertrand Russell, a genius at logic, was apparently not good at ontology, since he
referred to Leibniz's metaphysics as a “fairy tale” and unfairly and recklessly criticicized Meinong's theory of
objects in Russell's paper “On Denoting”. Russel claimed that:
a) Meinong's theory theory of obhjects was untenable, since everybody knows that objects are physical
(which is closed-minded, since Meinong's article was about mental as well as physical objects) and
b) Russell was promoting his own theory of descriptions, which purports to deal with descriptions of physical
objects (having assassinated mental objects), while physical objects. Russell should have learned from his own
text on Leibniz, are contingent. Although they maintain the same identity, they are constantly changing as their
perceptions and appetites change as well as their physical locations. Because of that, physical objects are constantly
hanging from their descriptions. In addition, as Descartes noted, solids can melt, but it is the same solid. Russell
hadn't read Descartes apparently. . And solids can burn and rust.
Russell was wrong on all counts, at least according to PL and common sense..
Thus it is no surprise that the philosophy of mathematics, in particular platonic mathematics,
terms such as “existence” and “abstract” which are not approprite to platonic iodealism, where all entities are in
idea-forms or ideas. Also, the PL used here combines and only combines the best features of Plato
(the One or Mind) with Leibniz's Many (the monads).
Having said all of that, and since the stanford site seems to partly confuse platonic realism with platonic
idealism (Leibniz is completely platonist idealism from the mental asspect) it seems important to state
some basic principles of a philosophy of mathematics which are in accord with Plato-Leibniz.
We begin with Leibniz's division of the ideal world into two parts, and our philosophy must be in accord with that
judgment, greatly simplifying it.

1. Necessary logic, which is always true, this being Plato's One or Mind. The only mathematical objects which might be
included here are numbers, theorems of arthmetic and Euclid's geometry,
including the Pythagorean theorem. This domain is timeless and space less except to include Euclid's geometry.
I believe this also include onstructions or operations themselves, allowing the square root itself, powers of numbers,
the equals sign, the not equal sign, greater than and lesser than. etc. But not an abstract statement such as y=x^(1/2),
or x>y, since x and y are abstractions. Nominalism applies.
to these objects since they are always true. Items in this catergory are also discovered, since necessary logic is always true,
and objects which are discovered, not invented, are part of Kant's a priori, which is another name for this category.
and

2. Contingent logic, which may not always be true. This allows any statement containing abstract entities
such as x and y, the equals sign, the not equal sign, greater than and lesser than. Thus this category
contains nearly the whole of mathematics and statements are to be considered theories or hypotheses
unless they are proven by experimental test.
In addition we note an unfortunate tendency in mathematics to attempt to apply traditionalal
linguistics such as language semiotics and language semantics to mathematics. This is fraught with
danger, as language—words-- are what Levi-Strauss called “bricolage” where meaning is obtained by
studying a pattern of different signs, and the meaning of each word, at least according to Wittgenstein,
is established pragmatically through usage in society. Wittgenstein's example of the simplest language is
by demonstration. You hand me an apple and say, “this is an apple.”



Dr. Roger B Clough NIST (retired, 2000).