Resources for Mathematics

Jon Barwise

Situations and Attitudes, (MIT 1983), coauthored by John Perry.

from p. ix:
In recent years a different perspective, one we associate with Thomas Reid, has been revitalized by the work of the psychologist J. J. Gibson and a famous paper by Hilary Putnam, "The meaning of 'Meaning.'" Gibson, studying the coordinated perception and action of animals, found much more information in the environment (and so less work to be done by the animal brain, or mind) than the traditional view of perception admitted; and Putnam's paper, we think, shows that there is much more meaning and information in the world and less in the head than the traditional view of meaning assumed. From these two complementary movements a new realism has emerged, Ecological Realism, a view that finds meaning located in the interaction of living things and their environment. What began for us as an exercise in technical philosohpy, reworking Frege, undoing his mistake, led us step by inevitable step to working out the beginnings of a theory of meaning implicit in this new realism.
Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena, (CSLI 1996), coauthored with Lawrence Moss.

from p. 5:
In certain circles, it has been thought that there is a conflict between circular phenomena, on the one hand, and mathematical rigor, on the other. This belief rests on two assumptions. One is that anything mathematically rigorous must be reducible to set theory. The other assumption is that the only coherent conception of set precludes circularity. As a result of these two assumptions, it is not uncommon to hear circular analyses of philosophical, linguistic, or comutational phenomena attacked on the grounds that they conflict with one of the basic axioms of mathematics. But both assumptions are mistaken and the attack is groundless.
The Logic of Distributed Systems, (Cambridge 1997), coauthored by Jerry Seligman.

from p. 219:
We have required that our regimentations live in the field of real numbers. An alternative suggestion would allow them to live in non-Archimedian fields, like the fields of nonstandard real numbers. One could then allow the tolerance to be infinitesimal, and sorites numbers could be infinite. This goes along with the discovery in recent years that nonstandard analysis often gives an elegant way to model the differences between the very small and the very large.

Shaughan Lavine

Understanding the Infinite (Harvard 1994)

from p. 133:
When combined with Skolem's arguments, those of von Neumann amount to a devastating criticism of our present-day axiomatic foundations of mathematics on the basis of set theory: Those foundations rely on a notion of proof, which requires a notion of finitude for its definition. But, once we specify the notion of definiteness, our axioms enable us to show that the definition of finitude they provide is an iniadequate foundation for the notion of proof. ... The criticism is not directed at the practice of using proofs, a practice that we can certainly acquire without having a theory of proofs or an adequate characterization of finitude. In practice all we need in the way of a theory of finitude is the recognition that any proof we actually encounter in completed form is finite, which is very far from a complete characterization of what it is to be finite. The criticism is directed at a certain kind of attempt to characterize or define what the practice allows in the way of proofs. The attempt fails.

Raymond Smullyan

Smullyan relates the puzzles confronted in mathematical logic to the broader scope of living.

5000 B.C. and Other Philosophical Fantasies
Lady or the Tiger? And Other Logic Puzzles Including a Mathematical Novel That Features Godel's Great Discovery
The Tao Is Silent
This Book Needs No Title : A Budget of Living Paradoxes, (Prentice Hall 1980, Simon & Schuster 1986)

from pp. 97-98:
"Are human beings like mountains and streams?" This is a profound question, and there seems to be remarkably divergent opinion on the matter! I think may people would say, "Of course they are different!" Expecially those brought up in a Jewish or Christian tradition would say: "Humans are alive. They have souls. They are rational. They have free will and make choices. They are capable of good and evil; they have moral responsibility; they are capable of sinning," etc., tec. On the other hand, there are those ultra-mechanistic types - often in the computer sciences - who say that human beings are basically like mountains and streams; they are both causally determined mechanisms. The human being (they say) may be a more complicated kind of machine but is a machine nevertheless and subject to exactly the same physical laws which govern the behavior of mountains and streams. Therefore human beings are like mountains and streams.
But what a ghastly, perverted way of looking at it! I don't know which of these two viewpoints I find more horrible! The first empasizes that aspect of religion I have always hated. The second is completely inhuman. I also tend to think of humans - at their best - as being like mountains and streams, but for such different reasons! In the first place, I don't think of humans as machines. Second, I don't think of the universe as a mchine; I think of it as an organism rather than a mechanism. Even if it is perfectly decribable, using purely mechanical and electromagnetic laws, I still think of it as an organism rather than a mechanism. And I think of mountains and streams as part of the organism of the cosmos, and a very beautiful part of that. And since I like to think of human beings as also beautiful, I therefore like to think of them - at their best - as being like mountains and streams.
What Is the Name of This Book? : The Riddle of Dracula and Other Logical Puzzles