In Science Without Numbers by Hartry Field he talks about stating things nominalistically. What seemed obvious to me is "How would you find radioactive half-life without math?". I looked all around and all methods to compute radioactive half-life all involved math. So here it is: how do you compute radioactive half-life in the method of Hartry Field(attractively nominalistically).


3 Answers 3


One way to express the concept of half-life without math is to fill a see-through container with pennies, marking their volume on the side with a horizontal line. Shake the container and then spread them out on a flat surface, removing all pennies that landed tails-up. Then return the remaining heads-up pennies back to the container. Mark the volume again (it should be lower). That's the first 'event'.

Repeat the same identical procedure with succeeding 'events', marking the changed volume each time, until there are no more pennies left. The side of your container should have several horizontal lines positioned one above the other. Observe the distances between the lines. Are they regular? Irregular? Graduated?

Next observe the line at the midpoint, approximately equidistant between full and empty (just eyeball it), representing the concept of "half-life". Compare the number of lines above it with those below that point. Are there more lines above it than below? Less? Or the about the same?

Use pennies to simulate unstable isotopes. When the pennies are heads up, they are radioactive (unstable). When the pennies are tails up, they have decayed to become stable isotopes.

How can you simulate radioactive half-life? (Socratic.org)


You don't. But that is not a defense of Platonism.

You cannot nominalistically determine that the pattern of decaying along an exponential curve exists in a way that meet's Fields' criterion of conservative derivation. So you cannot begin to calculate a number that describes the parameter for a given curve relevant to a given instance of decay. You have to bring the mathematical concept to the facts first, and then you can look at ways to instantiate its reference to your case.

The problem with the underlying theory here is that it does not provide a way of learning nominalistically. It is fully acceptable that mathematics is fictional, but, as Hume pointed out forever ago all observed patterns are fictional. They inject abstraction that is not even real most of the time, and even then is more than is warranted. The exact rate after the half-life averages out to half the original rate, but in almost no instance is it going to be exactly half. What puts these events into a pattern is the fact there is an average, a fiction we create for our own convenience. The notion of a nominalistic pattern is approximation to some mathematical proposal that we bring to the problems, from within our minds, because it works. With just the nominalism and no mathematics, there is no capacity to describe the pattern.

In the absolute, real world there are only events, not relations between events. Humans observe relations because we have developed the technology of the story, and we abstract those patterns into concise, separable concepts because we want to share them and to verify that other humans share them. All the patterns we observe are predictive guesses imposed on the world, which have content because they are symbolic. They can be transferred between people because they activate related imaginary content that is at some level shared, and otherwise negotiated.

There is no nominalistic science because science is about patterns, you cannot really name nonexistent objects, you can only name their interpretations as rulesets and the related expectations. I can name unicorns, but a unicorn is not a thing, it is a set of rules about something, and if a thing obeys those rules, it is likely to be perceived by others as a unicorn. I can name a person, or a place, or a rock. But the concept of a rock is not a thing that can be named until it is assembled as a pattern: a set of rules, which is made of other names, held together by our shared notion of logical combination, which is mathematics.

Those patterns don't have to pre-exist. They can be created at need. And they do not constitute a separate reality because they are in fact not real. They are artifacts of our potential for language, which is the possibility of fiction. When your dog learns to respond to the word 'food', it is because he has the capacity to generate fictional food, to have the same reaction to a symbol (in this case, of sound) that he would have to the presence of actual nourishment. That does not make the imaginary food real.

Popper's notion of falsifiability, and therefore active skepticism, as the basis of science, captures something real. We do science by bringing our biases to reality and having their usefulness disproved until we find the set that is hardest to refute. We are the authors of the fiction that becomes our descriptive context. But that does not keep it from being mere creative content.


Hartry Field's nominalism claims that mathematics is not merely consistent and convenient to use in calculations, but it is also "conservative". He describes this conservativeness property in this way (page x):

...any inference from nominalistic premises to a nominalistic conclusion that can be made with the help of mathematics could be made (usually more long-windedly) without it. This is a fundamental difference between the use of mathematical entities and the use of the theoretical entities of science: no such conservativeness property holds for the latter. The utility of theoretical entities in science is due solely to their theoretical indispensability: without theoretical entities, no (sufficiently attractive) theory is possible.

This would mean that one should be able to find a method to compute the half-life of radioactive elements without using math. Perhaps one method would be to place a certain amount of the material on a digital scale with a timer and collect weights and times until one reached half of the original weight.

There might be more practical ways to actually do that half-life calculation, but it did not involve the use of exponentials and logarithms to reach the result. They would have been more convenient to use, but they were not necessary.

Leon Horsten notes that this conservative property is controversial and Stewart Shapiro may have demonstrated that a counterexample should exist.

If the fictionalist thesis is correct, then one demand that must be imposed on mathematical theories is surely consistency. Yet Field adds to this a second requirement: mathematics must be conservative over natural science. This means, roughly, that whenever a statement of an empirical theory can be derived using mathematics, it can in principle also be derived without using any mathematical theories. If this were not the case, then an indispensability argument could be played out against fictionalism. Whether mathematics is in fact conservative over physics, for instance, is currently a matter of controversy. Shapiro has formulated an incompleteness argument that intends to refute Field’s claim.

The question asks how one might compute the half-life of a radioactive element. One way to do that would be to take measurements over time.

Hartry H. Field, Science Without Numbers: A Defense of Nominalism, 1980 Princeton.

Horsten, Leon, "Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Spring 2019 Edition), Edward N. Zalta (ed.), forthcoming URL = https://plato.stanford.edu/archives/spr2019/entries/philosophy-mathematics/.


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