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## The J. H. B. Archive report the Alexander Forbes papers

### Journal of the History of Biology (1978-09-01) 11: 387-393 , September 01, 1978

## Mathematics, the empirical facts, and logical necessity

### Erkenntnis (1983-05-01) 19: 167-192 , May 01, 1983

### Conclusion

We have argued that mathematical statements are *a posteriori synthetic* statements because they are ultimately based on *empirical* facts or on *empirical* hypotheses. In some cases, mathematical statements can be verified by direct inspection of an empirical model of the relevant mathematical structure. In other cases, they must be inferred from the axioms of the relevant axiomatic theories, on the assumption that these axioms are *consistent*. Yet, the consistency of these axioms is always an *empirical* assumption: it may admit of empirical *verification* (by exhibiting an empirical model for these axioms), or at least it will admit of potential empirical *falsification* (by deriving contradictory implications from them).

On the other hand, mathematical statements are a posteriori synthetic statements of a very special sort, because they do not depend on the *contingent* features of the empirical world, but rather are logically *necessary* properties of the mathematical structures realized or potentially realized in the empirical world (as shown by the invariance of these properties under isomorphism). Moreover, even though they are synthetic statements, they resemble analytic statements in being logical consequences of the *axiomatic definition* of the mathematical structure they are dealing with. For this reason, we have proposed the term *structure-analytic* statements to describe them.

Their logical status as structure-analytic statements gives them a logical position *intermediate* between truly analytic statements and ordinary empirical statements. This explains the nontrivial and nontautological character of many important mathematical theorems, which often gives them the quality of *a priori* quite unexpected “brute facts.” This is an aspect of mathematics very hard to explain on the logical positivist assumption that mathematical statements are *truly* analytic.

We have also discussed some of the philosophic and mathematical problems posed by various limitational theorems. We have argued that for Peano arithmetic the danger of inconsistency can be minimized (though it cannot be fully eliminated), and the problem of noncategoricity can be fully overcome, by stating it in the form of a quantifier-free recursive theory.

On the other hand, in the case of set theory, we have argued that the Skolem paradox shows that we are logically free to *reject* the existence of absolutely nondenumerable sets, yet that, both on intuitive and on pragmatic grounds, it is preferable to *admit* their existence, as most set theorists do.

Finally, we have found that we are logically free to opt *either* for dualism (or for pluralism) *or* for monism in set theory. If we opted for the former position, this would mean only that the theory of finite sets can be extended to infinite sets in two (or more) different but equally admissible ways-just as other mathematical theories can often be generalized in more ways than one. (Moreover, if infinite sets have the nature of ideal elements, as Hilbert has suggested, it cannot really surprise us if we find ourselves to be logically free to invent two or more different but equally consistent stories about them.) Indeed, we have argued, again both on intuitive and on pragmatic grounds, that it seems preferable to admit a need for *both* Cantorian and non-Cantorian set theories.

## Adaptive Contouring with Quadratic Tetrahedra

### Scientific Visualization: The Visual Extraction of Knowledge from Data (2006-01-01): 3-15 , January 01, 2006

### Summary

We present an algorithm for adaptively extracting and rendering isosurfaces of scalar-valued volume datasets represented by quadratic tetrahedra. Hierarchical tetrahedral meshes created by longest-edge bisection are used to construct a multiresolution *C*^{0}-continuous representation using quadratic basis functions. A new algorithm allows us to contour higher-order volume elements efficiently.

## Is genetic information irreducible?

### Biology and Philosophy (1996-10-01) 11: 535-538 , October 01, 1996

## Back Matter - Papers in Game Theory

### Papers in Game Theory (1982-01-01): 28 , January 01, 1982

## Are generic predictions enough?

### Erkenntnis (1989-03-01) 30: 43-68 , March 01, 1989

### Summary and conclusions

I have argued not that economics has no predictive content, but that it is limited, or at least has so far been limited to generic predictions. Now this is an important kind of prediction, and almost certainly a necessary preliminary to specific or quantitative predictions. But if the sketch of an important episode in the twentieth century history of the subject I have given is both correct and representative, then economics seems pretty well stuck at the level of generic prediction. And at least some influential economists and philosophers of economics seem well satisfied with stopping at the point of generic prediction. Or at least they give no other reason than its power to produce such predictions as a justification for the character of economic theory. But this leads to the question that is the title of my paper, is generic prediction enough?

## Generalized net structures of empirical theories. I

### Studia Logica (1977-09-01) 36: 195-211 , September 01, 1977

## Quantification, Modality, and Semantic Ascent

### Causality, Method, and Modality (1991-01-01) 48: 175-183 , January 01, 1991

Since Professor Vuillemin has studied several aspects of the thought of Bertrand Russell, it is perhaps appropriate here to juxtapose a program for metalinguistic interpretation of modality considered by Quine with a much more sweeping metalinguistic program earlier advocated by Russell.

## The politics and contexts of Soviet science studies (Naukovedenie): Soviet philosophy of science at the crossroads

### Studies in East European Thought (2011-06-29) 63: 175-202 , June 29, 2011

*Naukovedenie* (literarily meaning ‘science studies’), was first institutionalized in the Soviet Union in the twenties, then resurfaced and was widely publicized in the sixties, as a new mode of reflection on science, its history, its intellectual foundations, and its management, after which it dominated Soviet historiography of science until *perestroika*. Tracing the history of meta-studies of science in the USSR from its early institutionalization in the twenties when various political, theoretical and institutional struggles set the stage for the development of the field, to the sixties when the field resurfaced within the particular political context of the Cold War, and using the history of Moscow Institute for the History of Science and Technology as a case-study, I situate Soviet *naukovedenie* project within the culture of late-socialism in the Soviet Union during the Cold War, asking what this discourse meant for its creators and practitioners, as well as for the high-ranked Soviet officials who provided the authoritative support for this field.

## Neugenics?

### Monash Bioethics Review (2000-10-01) 19: 9-33 , October 01, 2000

Many are worried that the Human Genome Project will lead to a revival of eugenics. In this essay I examine the troublesome history of the ‘old eugenics’ which included the Nazi program of ‘Racial Hygiene’ and the sterilization of the ‘feebleminded’ in the United States of America. A ‘new eugenics’, involving prenatal diagnosis and the selective abortion of fetuses likely to develop into severely disabled infants, on the other hand, is claimed by many to be morally acceptable. If this is correct, then eugenics per se might not necessarily be an altogether bad thing. I therefore examine what was wrong with the old eugenics and what is often claimed to be different and better about the new eugenics. I argue that the morally relevant differences between the bad old eugenics and a potentially acceptable new eugenics are not accurately captured by a family of contrasts often thought to distinguish the two. I conclude that, in any case, the worry that the Human Genome Project will lead to a revival of eugenics is a red herring. The important thing is to realize that emerging genetic science and technology confronts both individuals and society as a whole with new kinds of reproductive decisions worthy of careful consideration.