# Author L. H. Kauffman

Biography: Louis H. Kauffman has a B.S. in mathematics from MIT and a PhD in mathematics from Princeton University. He is Professor of Mathematics at the University of Illinois at Chicago. Kauffman works on topology, knot theory, quantum topology, form and cybernetics. He is the founding editor and editorin- chief of the *Journal of Knot Theory and Its Ramifications* and the editor of the book series on *Knots and Everything*. He is the author of the column *Virtual Logic* for the journal *Cybernetics and Human Knowing*. Kauffman is the author of numerous books and papers on knot theory and is wellknown for his discovery of the bracket state sum model of the Jones polynomial and for his discovery of a generalization of the Jones polynomial called the Kauffman polynomial, for his discovery of virtual knot theory and other topological structures. He is the recipient of the 1993 Warren McCulloch award of the American Society for Cybernetics and the 1996 award of the Alternative Natuaral Philosophy Association for his work on discrete non-commutative electromagnetism. He was president of the American Society for Cybernetics from 2005 to 2008. Kauffman plays the clarinet in the Chicago-based ChickenFat Klezmer Orchestra.

Guddemi P., Brier S. & Kauffman L. H. (2015) Foreword: Ranulph Glanville and How to Live the Cybernetics of Unknowing A Festschrift Celebration of the Influence of a Researcher. Cybernetics & Human Knowing 22(2–3): 7–10.

Guddemi P., Brier S. & Kauffman L. H.
(

2015)

Foreword: Ranulph Glanville and How to Live the Cybernetics of Unknowing A Festschrift Celebration of the Influence of a Researcher.
Cybernetics & Human Knowing 22(2–3): 7–10.
Kauffman L. H. (1978) Network synthesis and Varela’s calculus. International Journal of General Systems 4: 179–187. Fulltext at http://cepa.info/1822

Kauffman L. H.
(

1978)

Network synthesis and Varela’s calculus.
International Journal of General Systems 4: 179–187.
Fulltext at http://cepa.info/1822
Network models are given for self-referential expressions in the calculus of indications (of G. Spencer Brown). A precise model is presented for the behavior of such expressions in time. The extension of Brown’s calculus by F. Varela is then shown to describe behavior invariant properties of these networks. Network design is discussed from this viewpoint.

Kauffman L. H. (1987) Imaginary values in mathematical logic. In: Proceedings of the Seventeenth International Conference on Multiple Valued Logic. IEEE: 282–289. Fulltext at http://cepa.info/1842

Kauffman L. H.
(

1987)

Imaginary values in mathematical logic.
In: Proceedings of the Seventeenth International Conference on Multiple Valued Logic. IEEE: 282–289.
Fulltext at http://cepa.info/1842
We discuss the relationship of G. Spencer-Brown’s Laws of Form with multiple-valued logic. The calculus of indications is presented as a diagrammatic formal system. This leads to new domains and values by allowing infinite and self-referential expressions that extend the system. We reformulate the Varela/Kauffman calculi for self-reference, and give a new completeness proof for the corresponding three-valued algebra (CSR).

Kauffman L. H. (1987) Self-reference and recursive forms. Journal of Social and Biological Structures 10: 53–72. Fulltext at http://cepa.info/1816

Kauffman L. H.
(

1987)

Self-reference and recursive forms.
Journal of Social and Biological Structures 10: 53–72.
Fulltext at http://cepa.info/1816
The purpose of this essay is to sketch a picture of the connections between the concept self-reference and important aspects of mathematical and cybernetic thinking. In order to accomplish this task, we begin with a very simple discussion of the meaning of self-reference and then let this unfold into many ideas. Not surprisingly, we encounter wholes and parts, distinctions, pointers and indications. local-global, circulation, feedback. recursion, invariance. self-similarity, re-entry of forms, paradox, and strange loops. But we also find topology, knots and weaves. fractal and recursive forms, infinity, curvature and imaginary numbers! A panoply of fundamental mathematical and physical ideas relating directly to the central turn of self-reference.

Kauffman L. H. (1998) Virtual logic – self-reference and the calculus of indications. Cybernetics & Human Knowing 5(2): 75–82.

Kauffman L. H.
(

1998)

Virtual logic – self-reference and the calculus of indications.
Cybernetics & Human Knowing 5(2): 75–82.
This is the fifth column in this series on “Virtual Logic.” In this column we shall begin by recalling the “Descent into the Form” that was described in the previous column, and how this descent makes clear how the mark of distinction can be seen as self-referential. We then relate this direct appearance of self-reference to other models and to Godel’s Incompleteness Theorem. The author apologizes beforehand for the ending of Part III of this essay with its modulation on the final lines of “Little Gidding” by T. S. Eliot.

Kauffman L. H. (1998) Virtual Logic – symbolic logic and the calculus of indications. Cybernetics & Human Knowing 5(3): 63–70. Fulltext at http://cepa.info/3114

Kauffman L. H.
(

1998)

Virtual Logic – symbolic logic and the calculus of indications.
Cybernetics & Human Knowing 5(3): 63–70.
Fulltext at http://cepa.info/3114
This is the sixth column in this series on “Virtual Logic.” In this column we shall give a short exposition of how symbolic logic is illuminated by the calculus of indications. Columns four and five began an introduction to the calculus of indications. Nevertheless, we shall be self-contained here and recall this construction in section 1.

Kauffman L. H. (1998) Virtual logic – The Smullyan Machine. Cybernetics & Human Knowing 5(4): 71–80. Fulltext at http://cepa.info/3119

Kauffman L. H.
(

1998)

Virtual logic – The Smullyan Machine.
Cybernetics & Human Knowing 5(4): 71–80.
Fulltext at http://cepa.info/3119
This is the seventh column in this series on “Virtual Logic.” In this column I will discuss an imaginary machine devised by the logician Raymond Smullyan. Smullyan managed to compress the essence of Gödel’s theorem on the incompleteness of formal systems into the properties of a devilish machine. This column consists in two parts. In the first part we find a story/satire about such a machine, with the Smullyan structure at its core. In this story, the protagonist is bent on detecting a flaw in the machine and he operates with strict two-valued logic. In such logic a statement is either true or false. Thus we call the statement “If unicorns can fly then all numbers are less than pi.” true because it is not definitely false. In general “A implies B” is taken to be false only if A is true and B is false. This is the one significant case where “A implies B” must be false. All other cases, such as A false and B true are taken to be true. This is the classical logical convention. It works quite well in its own domain, but it has its limits. One of these limits occurs when there is a gradation of qualities. For example in statements about tall and short the truth is relative to your idea of this discrimination. Another limit is in the realm of self-referential statements. Certainly the Liar Paradox – “This statement is false.” is neither true nor false in any timeless sense.

Kauffman L. H. (1999) Virtual logic – The Flagg Resolution. Cybernetics & Human Knowing 6(1): 87–96. Fulltext at http://cepa.info/3125

Kauffman L. H.
(

1999)

Virtual logic – The Flagg Resolution.
Cybernetics & Human Knowing 6(1): 87–96.
Fulltext at http://cepa.info/3125
This is my eighth column on virtual logic. In this column we shall consider a mode of paradox resolution that I call the “Flagg Resolution” after its inventor James M. Flagg.

Kauffman L. H. (1999) Virtual logic – The Matrix. Cybernetics & Human Knowing 6(3): 65–69. Fulltext at http://cepa.info/3135

Kauffman L. H.
(

1999)

Virtual logic – The Matrix.
Cybernetics & Human Knowing 6(3): 65–69.
Fulltext at http://cepa.info/3135
This is column number nine in the series. We take as our theme the recent movie “The Matrix.” The Matrix is a cinematic exercise in virtual reality and virtual logic. It is not necessary to have seen the film to read this essay. The Matrix is all around you. It is in the air you breathe, in the ground you walk on, in the sights you see and in the feelings that you feel. You yourself are composed of it just as much as it is composed of you. You imagine yourself to be an observer independent of the Matrix, but the very possibility of your observation, your sense of Self and World is produced by the Matrix.

Kauffman L. H. (1999) Virtual logic – The MetaGame Paradox. Cybernetics & Human Knowing 6(4): 73–79. Fulltext at http://cepa.info/3142

Kauffman L. H.
(

1999)

Virtual logic – The MetaGame Paradox.
Cybernetics & Human Knowing 6(4): 73–79.
Fulltext at http://cepa.info/3142
This is column number 10. In this column we shall discuss a recent relative [1], [2] of the Russell paradox, the Metagame Paradox. This paradox is related to a set theoretic paradox about well-founded sets, the Well-founded Set Paradox. These two paradoxes are both related to the basic nature of any observing system that would include itself in its own observations. I give you these paradoxes and a comment on the nature of the observer. Judge them for yourself. Near the end of the column we show how, by turning the paradox around, there are new proofs of uncountability of certain infinities. These proofs have the remarkable quality that they use a bit of imaginary reasoning, only to have it vanish just as it appears! It is this role of the imagination that is central to our theme. We reason by an imaginary detour. And we obtain a real answer.

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