Part 4' The Fly and the Fly Bottle

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Part 4. The Fly and the Fly Bottle

• Can an Inquiry into the Foundations
• of Mathematics Tell Us Anything
• Interesting about Mind?
• Written by Gabriel Stolzenberg
• Reviewed by Gary Berg-Cross
• Even mathematicians get trapped inside the fly
  bottle of dualistic thinking.

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Gabriel Stolzenberg
Gary Berg-Cross
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"Can an Inquiry into the Foundations of
Mathematics Tell Us Anything Interesting about
Mind?" by Gabriel Stolzenberg in P. Watzlawicks
Invented Reality How Do We Know What We Believe
We Know? (Contributions to constructivism).
Review by Gary Berg-Cross. Ph.D.
Engraving known as the Flammarion woodcut (1888)
depicting the Aristotelian conception of the
universe before the Copernican model - Pilgrim
peering through the sky as if it was a curtain,
after which you look at the hidden workings of
the universe..
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• Name Gary Berg-Cross Profession Cognitive
• Affiliation EMI Scientist
• Relation to Topic My PhD. Studied the
  construction of knowledge structures from
  Discourse.
• Question What are mathematical objects and is
  mathematics in a trap when it considers the
  foundations of mathematical knowledge? Is
  Mathematics Invented?
• Biologists think they are biochemists,
  Biochemists think they are Physical Chemists,
  Physical Chemists think they are Physicists,
  Physicists think they are Gods, And God
  thinks he is a Mathematician.
• (inference math is the ultimate truth.)  
• Mathematics is made of 50 percent formulas, 50
  percent proofs, and 50 percent imagination.
• Philosophy is a game with objectives and no rules
  whileMathematics is a game with rules and no
  objectives. 

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Alternative Views on Foundations of Mathematics
(FOM)

• Whats the Issue?
• Mathematical structure seem to mirror physical
  theory and often points the way to further
  advances in that theory and even to empirical
  predictions. So by induction we believe that this
  is not a coincidence but must reflect some larger
  and deeper truth about both mathematics and
  physics.
• Math is the foundation for the physical sciences.
  But what is maths foundation?
• Foundations of mathematics is a term sometimes
  used for certain fields of mathematics, such as
  mathematical logic, axiomatic set theory, proof
  theory, model theory, and recursion theory.
• The search for foundations of mathematics is also
  a central question of the philosophy of
  mathematics On what ultimate basis can
  mathematical statements be called true?
• Mathematical truth is demanded to have a
  rigidity, an absoluteness, that is tantamount to
  Platonism
• Platonist mathematical realism, seems exemplified
  by Kurt Gödel, who explored the existence of a
  world of mathematical objects independent of
  humans
• In this view the truths about these objects are
  discovered by humans.
• In this view, the laws of nature and the laws of
  mathematics have a similar status, and maths
  effectiveness ceases to be unreasonable.
• Not our axioms, but the very real world of
  mathematical objects forms the foundation. The
  obvious question, then, is how do we access this
  world?
• Alternate View
• The foundations of mathematics is, at least
  partly, a scientific study of mathematical
  practice.
• In this view what mathematicians actually do and
  actually say is of direct interest to the
  foundations of mathematics. We are Trapped into
  Believing otherwise.

See Wigners Unreasonable Effectiveness of
Mathematics in the Natural Sciences http//www.dar
tmouth.edu/matc/MathDrama/reading/Wigner.html
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An Inquiry Into the Foundations of Mathby
Gabriel Stolzenberg

• Intro View- the science of pure math fell into an
  intellectual trap
• Fixed habits of thought prevent us from create a
  closed system
• The Process of Entrapment
• The Concept of an Act of Acceptance as Such
• Acts in the Doman of Language use
• Story of a Definition that was too good to be
  true
• Belief Systems Attitudes about Undoing Accepted
  Belief
• Descriptive Fallacies Produced by a Failure to
  Respect Considerations of Standpoint
• The Case of Pure Mathematics Contemporary
  mathematicians Attachment to His Belief
• A View from the Edge of the System The Tug of
  Language that Pulls one Inside
• On Not Being Taken in by Language
• Statements to Signals Using Language to Make
  Knowledge Sharable
• Two mathematical Statements a Procedure for
  Getting into Position to make at Least One of
  Them
• Understanding Mathematical Statements
• One Way of making Mathematical Language Work
• The Contemporary Mathematicians Unfulfilled Task
  
• Making Mathematical Language function the Way he
  Wants it to
• A Pseudomystery about the nature of Mathematical
  Knowledge and the Manner in Which it is Acquired
• The Decisive Influence of Language Use on the
  Conduct of Mathematical Research

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The Process of Entrapment

• Since the late 19th century mathematicians with
  the help of logicians have been digging
  themselves deeply into a trap, but it is not
  perceived as that.
• The trap is built from certain structures of
  logic and language reflecting beliefs and
  habits of thought.
• Assumptions that are treated as givens/reality -
  reifications.
• Theres a role for proper scientific inquiry to
  help us out of a closer system, but most
  mathematicians arent asking the questions
• We need to step out of the trap to see it.

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The Concept of an Act of Acceptance as Such
Acts in the Doman of Language Use

• To accept something (experience or object) as
  such is to take it for what it appears (or is
  purported to be), and proceeding on that basis.
• In science we should explore the consequences of
  such assumptions. We should not easily
  accept/reify something that is an assumption.
• But in math the structures are built by ACTS
  which are accepted because they establish new
  theorems they become objects to manipulate.
• Suppose we find a problem with accepting a
  conclusion. Only then do we check the proof.
• But maybe what happened with Euclids 5th axiom
  (suppose IT is not TRUE) is more broadly
  applicable and this supposition testing should be
  the approach.
• Conclusion adopt an activist policy concerning
  the invention and following of procedures that
  entail the undoing of accepted belief and the
  habits of thought.
• A particular problem concerns language habits
• that make us accept some statements.

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Story of a Definition that was too good to be
true - PI

• We LEARN the DEFINITION but Its NOT obvious that
  the ratio of large small circles will have the
  same C/D ratio
• Proof should PRECEED definition in a well founded
  theory!
• Otherwise we have just knowledge of definitions
• We are victims of an education that teaches
  definitions w/o proof or checks
• Result is that our reasoning seems objective but
  is really standpoint oriented.

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Belief Systems Attitudes about Undoing Accepted
Belief

• We fall into traps from certain habits that
  include acceptance
• A natural motivation to have a world view
• But we can fall into the sin of certainty
  (Maturana) a belief system
• Therefore we need a method that doesnt yield to
  such habits of acceptance and untested belief
  thats scientific.
• If we accept a belief into our system, which
  then doesnt allow that belief to be challenged,
  then we are trapped.
• We might have been trapped by a formalist
  approach.
• To a Formalist, like Hilbert, mathematical
  theorems can be formulated as theorems of set
  theory.
• The truth of a mathematical statement, in this
  view, is then nothing but the claim that the
  statement can be derived from the axioms of set
  theory using the rules of formal logic.
• But in pure math the only criteria is consistency
  the consistency of the system itself. This may
  trap us in a System of thinking.
• As a counter we need adequate acceptance
  criteria.

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Descriptive Fallacies Produced by a Failure to
Respect Considerations of Standpoint

• Examples of Beliefs
• The sun is largely made of hydrogen
• Every mathematical statement is either T or F.
• Is this true or false?
• Does everyone see it this way? No we can have
  deviant logics modal,
• many-valued, dialethic, intuitionist, fuzzy etc.
• In 1951 Quines Two Dogmas Of Empiricism
  challenged received notions of knowledge, meaning
  and truth.
• He arguing that logic and maths, like factual
  statements, are open to revision in the light of
  experience.
• Experience, says Quine, does not confirm or
  falsify individual statements, but instead
  confronts an interlocking theory-laden system of
  statements, which has to be adjusted as a whole.
• And there cannot be any universally-held system
  of beliefs, he argued in his major work Word And
  Object (1960), since the way any theory describes
  the world is relative to that theory's linguistic
  background webs of belief.
• We use other criteria like simplicity and beauty
  when comparing systems and our logic SEEMS BETTER
  than the deviant ones when considering these.
• But even Quine found it difficult to break out of
  our classical logic system.
• Hes still an insider looking at the deviant
  outside from insider
• criteria.

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The Case of Pure Mathematics Contemporary
Mathematicians Attachment to His Belief

• Mathematicians are trapped by their shared math
  experiences (reality) in terms of math objects
  belonging to a set.
• This is learned and is seductively simpler than
  the deviant approaches, such as not accepted the
  law of excluded middle in the form that for any
  real numbers a, b, either a b or a ? b.
• The failure of these seemingly unquestionable
  principles in turn vitiates the proofs of a
  number of basic results of classical analysis.
• Most mathematicians consider the language of set
  theory foundational, simpler and more intuitive
  than that of lambda calculus (or the related
  category theory). So why switch?
• BUT we might reconsider if we ask, how does
  accepting the law of excluded middle contributes
  to the construction of what we call mathematical
  reality?

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A View from the Edge of the System The Tug of
Language that Pulls one Inside - Intuitionist
Logic

• Looking from the edge we have a better chance of
  freeing ourselves from being too inside a system.
  We can contrast inside and outside views by
  considering Goldbach's Conjecture, that all even
  numbers larger than 4 is the sum of two primes
  18 13 5, or 102 97 5.
• At the edge or our understanding taking an inside
  stance (realist perspective) it makes sense to
  suppose that this conjecture might be true
  because every one of the infinite series of even
  numbers exists as a sum or two primes, even
  though there might be NO proof to be discovered.
• To a realist the answer already exists, but is
  unknown to us, its real, a thingthat we
  discover.
• Our author GS follows Mike Dummetts approach
  (anti-realism). To understand this assertion
  means being capable of recognizing what would
  count as evidence for or against it.
• Evidence is based on a procedure.
• As far as the intuitionist/anti-realist is
  concerned, the only thing that COULD make the
  conjecture TRUE is that there be a proof, a
  procedure to establish it. We construct it.
  Before that it is indeterminate.
• For all we know, according to the intuitionist,
  there might be NO PROOF and no counter-example,
  in which case there is nothing to give the
  conjecture a truth-value.
• The belief that every proposition is
  determinately true or false is
• the principle of bivalence. Its an assumption.
• Dont accept that knowing an answer means
  something EXISTS

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On Not Being Taken in by Language and Statements
to Signals Using Language to Make Knowledge
Sharable

• Indeterminacy seems distasteful especially when
  each step we think of is determinant.
• Why this reaction to the idea that an answer may
  not exist before a procedure was carried out? GS
  proceeds as follows-
• Is it the Platonic conceptual habits of language?
  Discovery not Creation of answers" in
  statements. Answers need not be things.
• Consider that the role of language is to make
  knowledge sharable (Sq root of 2 is not a
  rational )
• We must already share an understanding of how
  language is used to share this new info.
• When we establish this initial understanding we
  may limit other understandings.
• Thus a statement in nothing more than a signal
  of being in possession of some piece of
  knowledge.
• To inquire about a statements veracity is
  independent of our knowing it. Statements arent
  things they are only acts of stating.

From Theories, Models, Reasoning,Language, and
Truthby John F. Sowa. See also Gary Berg-Cross.
A Pragmatic Approach to Discussing Intelligence
in Systems, PerMIS 2004 and my discussion of
Scruffy Vs. Neat Approaches Models in
Information Assimilation and Indexed Knowledge
by Gary Berg-Cross BCIG 2002
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Two mathematical Statements a Procedure for
Getting into Position to Make at Least One of Them

• An invented example (you can invent your own)
• The author, GS, considers the fractional part of
  the sequence (3/2)n and
• discusses an assertion on where they fall as n
  goes from 1 to 2,000,000.
• We can assert (S) that more than 1 million fall
  in the interval 0-1/2.
• Or we can assert (T) that more than 1 million
  fall in the interval interval ½ to 1.
• It seems clear why one or the other will be true
  procedurally.
• So is it discovery or creation? GS argues that
  it is better captured by a metaphor/sign of
  creation and not discovery.

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Understanding Mathematical Statements One Way
of Making Mathematical Language Work

• If you ask a philosopher what the main problems
  are in the philosophy of mathematics, then the
  following 2 are likely to come up
• what is the status of mathematical truth, and
  what is the nature of mathematical objects? That
  is, what gives mathematical statements their aura
  of infallibility, and
• what on earth are these statements about?
• W. T. Gowers, Does mathematics need a
  philosophy?, presented at Cambridge University
  Society for the Philosophy of Mathematics 2002
• Heres the confusion. The statements more than
  a million terms of the sequence", and
  fractional part etc. all are in present tense
  and thus seem to say that the terms already
  exist.
• Present tense is oriented to the world of
  objects.
• But all that is in the present tense is our
  signaling that we understand, which is a
  convention established in our language.
• And if we accept this view, then perhaps lack of
  existence is also true when we talk about
  number sets functions.
• As scientists we should not be uncritically using
  language like this
• In GSs view mathematical language should signal
  mathematical knowledge involves conventions
  that govern the use of mathematical language.
• Conventions are established by usto a purpose
  such as K or acts of strict counting and
  performing computation.
• E.g. conventions on what ABC mean for 57 and
  AB etc.
• The present tense in these is purely
  metaphorical.
• It means if I want to be in the state of
  knowing I have a procedure/computation to arrive
  at something that witnesses the validity of the
  assertion.

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The Contemporary Mathematicians Unfulfilled
Task Making Mathematical Language function the
Way He Wants it To

• Using a constructive method as convention, math
  proofs would be of genuine scientific use in the
  traditional sense the proofs do say something
  about acquiring knowledge.
• But the Contemporary Mathematician prefers to say
  that 347 is not about acts of counting but
  about the relation of numbers.
• Our author claims that we dont have proper
  conventions to really support this.
• What sort of thing is a set? We have
  intuitions but structures to model these seem
  insufficient to outsiders.
• Better to talk about operational interpretations
  of sets.

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A Pseudomystery about the nature of Mathematical
Knowledge and the Manner in Which it is Acquired

• Here we enter the faculty of mind issue which
  is a Pseudomystery, since the Mathematician cant
  explain Mathematical Knowledge and how it is
  Acquired.
• Mathematicians conclude that it tells us about
  limits of our mind.
• Both S and its negation are unprovable,
• The First Incompleteness Theorem
• What Gödel points out is that if a formal logical
  system contains a statement S, "This statement is
  unprovable" which can be proven true, then we
  have a contradiction.
• Because the statement is provable, despite claims
  to the contrary.
• On the other hand, if the statement is false,
  then the theory is incompleteit contains a
  statement which is not able to be proved.
• This means no formal logical system can be both
  consistent and complete.
• But this may tell us nothing really interesting
  about the mind only the incorrect assumptions
  about mathematical objects.
• Consider
• A great truth is a statement whose opposite is
  also a great truth. Niels Bohr
• We have to do mathematics using the brain which
  evolved 100,000 years ago for survival in the
  African savannah. Stanislav Dehaene

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The Decisive Influence of Language Use on the
Conduct of Mathematical Research The Influence
of Talk About Statements Being True

• Does this understanding of language problems etc.
  lead to a mathematics that is useful and a method
  that is practical in daily work?
• Yes we can reformulate mathematical statements to
  eliminate talk about mathematical objects.
• If, for each X, either statement A(x) or B(x) is
  true then either all statements A(x) are true or
  some statement B(x) is true.
• What is present tense is the statement and not an
  object referenced by the statement.
• As for everything else, so for a mathematical
  theory
• beauty can be perceived but not explained.
  Arthur Cayley

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Concluding Remarks - The Title Question Answered

• In mathematics how do we attain knowledge of
  object independent of us? Maybe
  we dont.
• Its more an Illusion, produced by acceptance of
  beliefs that is encouraged by language habits
  than a special mental faculty.
• Its not knowledge acquired in a special way, it
  is subject to scientific criteria, its
  empirical.
• Numbers are tools, not rules. 
• Numbers are symbols for things the number and
  the thing are not the same.
• Skill in manipulating numbers is a talent, not
  evidence of divine guidance.
• The product of an arithmetical computation is the
  answer to an equation it is not the solution to
  a problem.
• Arithmetical proofs of theorems that do not have
  arithmetical bases prove nothing.
  Ashley-Perry from
  Statistical Axioms

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Supplementary Material

• In case you want a bit more on this topic and
  related topics
• Gary Berg-Cross

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Our Author - Gabriel Stolzenberg

• Gabriel Stolzenberg is a Professor of Mathematics
  at Northeastern University. He has an A.B. from
  Columbia, a doctorate from MIT, and an honorary
  master's degree from Brown.
• He has been a fellow of the Alfred P. Sloan
  Foundation and the John Simon Guggenheim
  Foundation.
• He spent one year at the Institut des Hautes
  Etudes Scientifiques in France.
• He has been a Visiting Professor at the
  University of Paris, Orsay, the University of
  Caifornia at San Diego and MIT.
• His best known mathematical work is "Volumes,
  Limits and Extensions of Analytic Varieties"
  (Lecture Notes in Mathematics, No. 19, Springer
  1966).
• His best known works about mathematics are
• his critical review of Errett Bishop's,
  "Foundations of Constructive Analysis" (Bulletin
  of the American Mathematical Society, March 1970)
  and
• "Can an Inquiry into the Foundations of
  Mathematics Tell Us Anything Interesting About
  Mind?" (in Psychology and Biology of Language and
  Thought Essays in Honor of Eric Lenneberg,
  edited by Eliza Lenneberg and George A. Miller,
  Academic Press, 1978).
• His non-mathematical interests include
  metaphysics, belief formation, interpretation of
  texts and legal theory. His main research has
  been on constructivist mathematics and the
  classical/constructivist gestalt shift.

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Some References

• Errett Bishop, Foundations of constructive
  analysis (1967)
• L.E.J. Brouwer, Collected Works, Vol. 1
• Alonzo Church, "An Unsolvable Problem of
  Elementary Number Theory," American Journal of
  Mathematics, Vol. 58, 1936, p. 351, fn. 10 --
  also in The Undecidable, ed. Martin Davis (1965),
  pp. 88-107
• Michael Dummett, Elements of Intuitionism (1977),
  
• von Glasersfeld, E. (1974). Piaget and the
  radical constructivist epistemology. In C. D.
  Smock E. von Glasersfeld (Eds.), Epistemology
  and education. Athens, GA Follow Through
  Publications. Alfred Tarski, Logic, Semantics,
  Metamathematics (1956)
• Quine, W. V. Philosophy of Logic. 2d ed.
  Cambridge, MA Harvard University Press, 1986.
• Alfred Tarski, "The Semantic Conception of
  Truth," Philosophy and Phenomenological Research
  (1944), pp. 347, 349, 355, etc.
• Ludwig Wittgenstein, Philosophical Grammar (1974)
  (although not mentioned in the chapter, he Like
  Kuhn seems relevant Gary Berg-Cross
• Ludwig Wittgenstein, Wittgenstein's Lectures on
  the Foundations of Mathematics (1976)

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More on Hilberts Influence

• German mathematician who reduced geometry to a
  series of axioms and contributed substantially to
  the establishment of Formalism in foundations of
  mathematics.
• His work in 1909 on integral equations led to
  20th-century research in functional analysis his
  Formalism, which treats mathematics as a series
  of symbols which can be rearranged according to
  various formal rules, sheds no light on the
  obvious relevance of mathematics to natural
  science, but led to the development of
  meta-mathematics, the systematic study of the
  comparative structures of mathematical theories,
  which was essential to subsequent elucidation of
  the problems of the foundations of mathematics by
  Gödel and others.
• Hilbert completed his PhD at the University of
  Königsberg in 1884, remaining there till 1895,
  after which he was appointed Professor of
  Mathematics at the University of Göttingen, where
  he remained for the rest of his life.
• Hilbert introduced a highly original approach to
  the consideration of mathematical invariants
  which allowed the structure of mathematical
  theories to be themselves the subject of
  mathematical analysis in a way hither to
  unimagined.
• An invariant is that aspect of something which
  remains the same when a corresponding
  transformation is applied to something.
• The meaning of invariant is most easily
  understood in terms of geometrical
  transformations such as displacement (where shape
  and size are invariant) or dilation (where shape
  but not size are invariant).
• Hilbert proved that all invariants can be
  expressed in terms of a finite number. His new
  method allowed him to produce a set of axioms for
  Euclidean geometry which marked a turning point
  in the theory of axioms.

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Foundations for Classical Logic
Classical logic rests upon 3 axioms. These axioms
are held to be 'self evident'because all
syllogisms rely on them, and because they can be
defended through retortion. "Retortion" means
that any syllogism used to refute these axioms
will have to rely on them (!!!!!) - leading to a
self refutation (we call this type of self
refutation the "Stolen concept fallacy"). You
can see why GS talks about being trapped in a
system!!!
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What are Watzlawicks Roots on Invented Reality?
Constructionism.
He was associated with Gregory Bateson who was
influenced by Korzybski Number is different
from quantity. Gregory Bateson
The map is not the territory, and the name is
not the thing named. Alfred Korzybski.
He was part of the Cybernetics era. The
anthropologists Gregory Bateson and Margaret Mead
contrasted first and second-order Cybernetics in
the diagram on the right (1973). It emphasizes
the requirement for a possibly constructivist
participant observer in the second order case as
shown in their figure.
Second-order cybernetics was created from the
attempts of classical cyberneticians to construct
a model of the mind. Researchers realized
that . . . a brain is required to write a theory
of a brain. From this follows that a theory of
the brain, that has any aspirations for
completeness, has to account for the writing of
this theory. And even more fascinating, the
writer of this theory has to account for her or
himself. Translated into the domain of
cybernetics the cybernetician, by entering his
own domain, has to account for his or her own
activity. Cybernetics then becomes cybernetics of
cybernetics, or second-order cybernetics. Heinz
von Foerster
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General Systems and Constructionists
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Other Critiques of mathematical Foundations

• Karl Popper believed that mathematics was not
  experimentally falsifiable and thus NOT a
  science.
• However, other philosophers, e.g. Imre Lakatos,
  have applied a version of falsificationism to
  mathematics itself.
• An alternative view is that certain scientific
  fields (such as theoretical physics) are
  mathematics with axioms that are intended to
  correspond to reality.
• The theoretical physicist, J. M. Ziman, proposed
  that scienctific knowledge is public knowledge
  and thus includes mathematics.?
• What mathematics seems to share with many of the
  physical sciences is the exploration of the
  logical consequences of assumptions.
• Intuition and experimentation also play a role
  in the formulation of conjectures needed grow
  both mathematics and the (other) sciences.
• Experimental mathematics continues to grow in
  importance within mathematics, and computation
  and simulation are playing an increasing role in
  both the sciences and mathematics, weakening the
  objection that mathematics does not utilize the
  scientific method.
• In his 2002 book A New Kind of Science, Stephen
  Wolfram argues for a new kind of computational
  mathematics deserves to be explored empirically
  as a scientific field in its own right. Maybe it
  is the real math.

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Some Related Publications by Gary Berg-Cross

• Ph.D. SUNY Stony Brook, Cognitive Psychologist.
  1974
• Dissertation Semantic model and organization of
  knowledge for discourse
• BS Rensselaer Polytechnic Institute,
  Experimental Psychology, 1967
• Relevant Publications
• Incremental Semantics for Service Oriented
  Architecture Presented at SOA EGov Conference,
  at Mitre, Falls Church VA May 2007
• Developing Knowledge for Intelligent Agents
  Exploring Parallels in Ontological Analysis and
  Epigenetic Robotics, NIST PerMIS conferences 2006
• A Pragmatic Approach to Discussing Intelligence
  in Systems, PerMIS 2004
• Dimensions of Intelligence and Intelligent
  Systems at PerMIS conference, August, 2002,
• Applying an Ontological Analysis Methods and
  Ontological Design Patterns in Support of the EHR
  , Poster at KR-MED's "Biomedical Ontology in
  Action" workshop, Nov. 8, 2006
• Handling Adaptive Complexity through
  Intermediate Developmental Cognitive Structures
  Applications of Cognitive Complexity in
  Organizations paper at ECHO workshop on
  Inquiries, Indices and Incommensurabilities ,
  GWU, Sept, 2004.
• Exploring eGov Cooperation and Knowledge Sharing,
  presented at Toward More Transparent Government.
  Workshop on eGovernment and the Web, sponsored by
  W3C, held 18 - 19 June at the National Academy of
  Sciences
• (with, John Hanna) Using Conceptual Structures
  to Translate Data Models Concepts, Context and
  Cognitive Processes. Workshop on Conceptual
  Graphs 1992 171-187
• (with, John Hanna) Can a large knowledge base be
  built by importing and unifying diverse
  knowledge? lessons from scruffy work.
  Knowl.-Based Syst. 5(3) 245-254 (1992)

Gary Berg-Cross with daughter Amber grandson
Caden