Nelson Goodman

The calculus of individuals and its uses

An individual or whole we understand to be whatever is represented in any given discourse by signs belonging to the lowest logical type of which that discourse makes use. What is conceived as an individual and what as a class is thus relative to the discourse within which the conception occurs. One task of applied logic is to determine which entities are to be construed as individuals and which as classes when the purpose is the development of a comprehensive systematic discourse. The concept of an individual and that of a class may be regarded as different devices for distinguishing one segment of the total universe from all that remains. In both cases, the differentiated segment is potentially divisible, and may even be physically discontinuous. The difference in the concepts lies in this: that to conceive a segment as a whole or individual offers no suggestion as to what these subdivisions, if any, must be, whereas to conceive a segment as a class imposes a definite scheme of subdivision—into subclasses and members. The relations of segments of the universe are treated in traditional logistic at two places, first in its theorems concerning the identity and diversity of individuals, and second in its calculus of membership and class-inclusion. But further relations of segments and of classes frequently demand consideration. For example, what is the relation of the class of windows to the class of buildings? No member of either class is a member of the other, nor are any of the segments isolated by the one concept identical with segments isolated by the other. Yet the classes themselves have a very definite relation in that each window is a part of some building. We cannot express this fact in the language of a logistic which lacks a part-whole relation between individuals unless, by making use of some special physical theory, we raise the logical type of each window and each building to the level of a class—say a class of atoms—such that any class of atoms that is a window will be included (class-inclusion) in some class that is a building. Such an unfortunate dependence of logical formulation upon the discovery and adoption of a special physical theory, or even upon the presumption that such a suitable theory could in every case be discovered in the course of time, indicates serious deficiencies in the ordinary logistic. Furthermore, a raising of type like that illustrated above is often precluded in a constructional system by other considerations governing the choice of primitive ideas.

Steps Toward a Constructive Nominalism

We do not believe in abstract entities. No one supposes that abstract entities—classes, relations, properties, etc.— exist in space-time; but we mean more than this. We renounce them altogether. We shall not forego all use of predicates and other words that are often taken to name abstract objects. We may still write ‘ x is a dog,’ or ‘ x is between y and z ’; for here ‘is a dog’ and ‘is between … and’ can be construed as syncate-gorematic: significant in context but naming nothing. But we cannot use variables that call for abstract objects as values. In ‘ x is a dog,’ only concrete objects are appropriate values of the variable. In contrast, the variable in ‘ x is a zoological species’ calls for abstract objects as values (unless, of course, we can somehow identify the various zoological species with certain concrete objects). Any system that countenances abstract entities we deem unsatisfactory as a final philosophy.

Propionic acid fermentation of ultra-high-temperature sterilized whey using mono-and mixed-cultures

SummaryUnlike sterilization by autoclave (Anderson et al. 1986) high concentrations of cheese whey sterilized by ultra high temperature (UHT) resulted in a medium conducive to microbial growth and propionic acid production. Propionibacterium freudenreichii ss. shermanii, grown with pH control in 12% whey solids and 1% yeast extract sterilized by UHT, produced about 1.9% propionic acid within 70 h; more than 50% of the lactose was not used. Under similar conditions, mixed cultures of P. shermanii and Lactobacillus casei produced more than 3.0% propionic acid. Acclimating the mixed culture to the whey medium resulted in 4.5% propionic acid. The amount of propionic acid produced was further increased to about 6.5% by raising the concentration of whey solids to about 18%. Using the mixed culture, all the lactose was consumed and lactic acid did not accumulate.

Safety, Strength, Simplicity

When the evidence leaves us with a choice among hypotheses of unequal strength, how is the choice to be made? Caution would counsel us to choose the weakest, the hypothesis that asserts the least, since it is the least likely to fail us later. But the principle of maximum safety quickly reduces to absurdity; for it always dictates the choice of a hypothesis that does not go beyond the evidence at all.

Twisted Tales; Or, Story, Study, and Symphony

Nothing strikes us as unusual here even though the order of telling completely reverses the order of occurrence. Indeed, to have withheld the result of the race to the end would have been inconsiderate under the circumstances. In other reports the telling may jump back and forth; for example: Excalibur, though he broke last from the gate, won by a nose after having dropped back to fourth, coming into the stretch, from the lead he had taken by the far turn.

Routes of Reference

Routes of reference are quite independent of roots of reference. I am concerned here with the various relationships that may obtain between a term or other sign or symbol and what it refers to, not with how such relationships are established. And since I am thus concerned with structures rather than origins, I shall not be discussing such topics as speechact or so-called causal theories of reference. My subject is the nature and varieties of reference regardless of how or when or why or by whom that reference is effected.

On the simplicity of ideas

The motives for seeking economy in the basis of a system are much the same as the motives for constructing the system itself. A given idea A need be left as primitive in a system only so long as we have discovered between A and the other primitives no relationship intimate enough to permit defining A in terms of them; hence the more the set of primitives can be reduced without becoming inadequate, the more comprehensively will the system exhibit the network of interrelationships that comprise its subject-matter. Of course we are often concerned less with an explicit effort to reduce our basis than with particular problems as to how to define certain ideas from others. But such special problems of derivation, such problems of rendering certain ideas eliminable in favor of others, are merely instances of the general problem of economy. Thus it is quite wrong to think of the search for economy as a sort of game, inspired by an abnormal love of superficial neatness. Some economies may be relatively unimportant, but the inevitable result of regarding all economy as trivial would be a willingness to accept all ideas as primitive at the outset, making a system both unnecessary and impossible.

Interpretation and Identity: Can the Work Survive the World?

Predictions concerning the end of the world have proven even less reliable than your brokers recommendations or your fondest hopes. Whether you await the end fearfully or eagerly, you may rest assured that it will never come-not because the world is everlasting but because it has already ended, if indeed it ever began. But we need not mourn, for the world is indeed well lost, and with it the stultifying stereotypes of absolutism: the absurd notions of science as the effort to discover a unique, prepackaged, but unfortunately undiscoverable reality, and of truth as agreement with that inaccessible reality. All notions of pure givenness and unconditional necessity and of a single correct perspective and system of categories are lost as well.

Production of propionic acid by mixed cultures ofPropionibacterium shermanii andLactobacillus casei in autoclave-sterilized whey

SummaryPure cultures ofPropionibacterium freudenreichii ss.shermanii did not grow in autoclave-sterilized cheese whey (121°C, 15 psi, 20 min) at whey concentrations greater than 2% (w/v) spray-dried sweet dairy whey. Propionic acid was produced from autoclave-sterilized whey by growingP. shermanii in mixed culture withLactobacillus casei. In medium containing 5–12% autoclaved whey solids and 1% yeast extract, the mixed culture produced 1.3–3.0% propionic acid, 0.5–1.0% acetic acid, and 0.05–0.80% lactic acid. All the lactose was consumed. Using pH-controlled fermentors (pH=7.0), mixed cultures produced at least 30% more propionic acid than cultures in which pH was not controlled.

Approach to the Problems

As an introduction to the detailed study of various systems, I shall in the present chapter attempt to clarify briefly some aspects of their common subject matter, characterize certain of the more important types of basis that may be chosen, and introduce some of the first problems that are to be dealt with. Much of this preliminary discussion will be quite informal, making free use of such everyday language as seems helpful and often dealing rather summarily with terms (such as “thing”) whose formal definition we shall not reach at all in this book.