Axioms in science and the scientific method

Question

Do axioms exist in the scientific method like in physics, chemistry, biology, …? E.g. Ockham's Razor for picking the best theory – is it an axiom? Another example is that science gathers data, analyzes it and comes to a conclusion through reasoning (usually through inductive reasoning)… but can these axioms change in the future if we find a better way? Or would they be forever set (because they are axioms)?

Answer 1

Yes, axioms do exist. Underlying the processes of science are several philosophical assumptions--aka 'axioms' or 'first principles.' They are necessary for making any and all inferences from scientific data, and really, even for the application and method of science itself. We take them for granted--like most philosophy--and don't think about them much. They are unspoken but very present and real. They are a baseline we cannot go beyond which means they provide the foundation for all empirical science while being empirically unprovable themselves.

The principle of contradiction, of the excluded middle, of noncontradiction, of identity, of intelligibility, of sufficient reason, of causal closure, of finality, and at least one principle of substance, are all ‘first principles’ that have been demonstrated through application after being used, but which cannot, themselves, be anything but assumed a priori.

We must assume the universe is a basically rational place; that cause and effect are rationally predictable at least with probability; and that knowledge from the past provides a rational foundation for studying the present and predicting the future. In other words, we must assume the laws of causality and the uniformity of nature for science and also for knowledge itself.

Hume wrote on the uniformity of nature demonstrating its unreliability and its unavoidable nature as an assumption at the same time, but without it, most of the past would be undecipherable to us.

Insofar as anyone can tell, our universe is orderly and logical. This is an inference made using reductionism, which assumes that studying a part of something will produce real knowledge of the whole. Reductionism represents a certain perspective on causality, and while it is not a universally supported assumption, (since some claim reductionism produces a fragmented picture of reality, when reality is actually composed of wholes), it is a necessary assumption for most of the day-to-day work of science. Astrophysicists remain open to finding alternative answers about the universe itself, of course, but reductionism remains one of the three most basic--and useful--foundational assumptions of all science.

All of this assumes reason, and experience, are dependable sources of knowledge. People are likely to breeze across this one with a snort and “of course they are!” However, what most assume is the dependability of their own experiences; they do not assume the dependability of someone else’s. Hence the response, “I don’t care what your experience is, it doesn’t line up with mine, therefore you are clearly deluded.” Some experiences can be misleading, yet there we are-—we have to assume reason and experience are dependable ways to knowing or we have no way to know anything at all.

We must assume our minds are capable of understanding nature. 'Truths' may be out there, but if we can’t discern them, how can we know? Science must assume it is possible to know, to discover—-not to create-—but to find, real knowledge and understanding or there would be no science at all.

There can be no knowledge without a 'knowing agent', therefore this assumes the existence of a “mind” and a “self” that is capable of interacting with a real world around it. This has become an increasingly argued assumption due to Nick Bostrom's simulation hypothesis trilemma--which serves to demonstrate that it has been an assumption.

Science consists of taking measurements or conducting experiments, and the data from those help scientists develop new hypotheses and theories. This assumes error is correctable with new data, and that knowledge is attainable; without that science stops. This assumes some kind of ‘truth’ is a real thing — that truth can be found and error shown and identified and corrected. This assumes that not all knowledge is equal.

This is an extensive but not an exhaustive list.

Yes axioms exist in science. They are the foundation of all empirical reasoning, but, as they are not founded on empiricism, they are not falsifiable, so they generally don't change much. But they can be challenged. There are a few that are currently undergoing significant challenges. Causal closure--the assumption that everything physical has a physical cause--for example, is currently undergoing a significant challenge from philosophy of mind advocates. Reductionism has been and continues to face challenges. The very nature of our reality and perception are being challenged by Bostrom and his supporters.

There are always challenges, but axioms remain in one form or another.

http://aynrandlexicon.com/lexicon/causality.html http://www.nyu.edu/gsas/dept/philo/courses/modern05/Hume_on_empirical_reasoning.pdf https://books.google.com/books?id=WuD8yaYxv-wC&printsec=frontcover&dq=Self-Knowing+Agents,+by+Lucy+O%E2%80%99Brien.&hl=en&ppis=_c&sa=X&ved=2ahUKEwj7scLd6pfnAhXSmeAKHf9lAuYQ6AEwAHoECAQQAg#v=onepage&q=Self-Knowing%20Agents%2C%20by%20Lucy%20O%E2%80%99Brien.&f=false https://www.iep.utm.edu/red-ism/ https://academic.oup.com/pq/article-abstract/65/261/626/1506037?redirectedFrom=fulltext https://www.jstor.org/stable/3751725?read-now=1&refreqid=excelsior%3Aed50644791272fdb31afc0204f3cde52&seq=1#page_scan_tab_contents

Answer 2

cf. Hilbert's 6^th problem: "Can physics be axiomized?" of his Mathematics Problems lecture delivered before the International Congress of Mathematicians at Paris in 1900:

6. Mathematical treatment of the axioms of physics

The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.

As to the axioms of the theory of probabilities,¹⁴ it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases.

Important investigations by physicists on the foundations of mechanics are at hand; I refer to the writings of Mach,¹⁵ Hertz,¹⁶ Boltzmann¹⁷ and Volkmann.¹⁸ It is therefore very desirable that the discussion of the foundations of mechanics be taken up by mathematicians also. Thus Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua. Conversely one might try to derive the laws of the motion of rigid bodies by a limiting process from a system of axioms depending upon the idea of continuously varying conditions of a material filling all space continuously, these conditions being defined by parameters. For the question as to the equivalence of different systems of axioms is always of great theoretical interest.

If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories. At the same time Lie's a principle of subdivision can perhaps be derived from profound theory of infinite transformation groups. The mathematician will have also to take account not only of those theories coming near to reality, but also, as in geometry, of all logically possible theories. He must be always alert to obtain a complete survey of all conclusions derivable from the system of axioms assumed.

Further, the mathematician has the duty to test exactly in each instance whether the new axioms are compatible with the previous ones. The physicist, as his theories develop, often finds himself forced by the results of his experiments to make new hypotheses, while he depends, with respect to the compatibility of the new hypotheses with the old axioms, solely upon these experiments or upon a certain physical intuition, a practice which in the rigorously logical building up of a theory is not admissible. The desired proof of the compatibility of all assumptions seems to me also of importance, because the effort to obtain such proof always forces us most effectually to an exact formulation of the axioms.

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<sup>14. Cf. G. Bohlmann, "Ueber Versicherungsmathematik," from the collection: F. Klein and E. Riecke, Ueber angewandte Mathematik und Physik, Teubner, Leipzig, 1900.
15. E. Mach: Die Mechanik in ihrer Entwickelnng, Brockhaus, Leipzig, 4th edition, 1901.
16. H. Hertz: Die Prinzipien der Mechanik, Leipzig, 1894.
17. L. Boltzmann: Vorlesungen über die Principe der Mechanik, Leipzig, 1897.
18. P. Volkmann: Einführung in das Studium der theoretischen Physik, Teubner, Leipzig, 1900.</sup>

Answer 3

The short answer:

1. There is no such thing as the scientific method.
2. Axioms exist within theories and are called postulates. However, they don't typically translate across theories.
3. Ochman's Razor is not an axiom or postulate, but rather a guideline for pick what theory to believe when you have two or more competing theories that explain the data equally well.

Answer 4

It appears that you're presupposing that the axiomatic method in mathematics is the essence of mathematics and given its success in coming up with irrefutable truths one should search for ways of axiomatising other sciences; but this presupposition may not hold; here's Poincares take on this in his book Science & Hypothesis:

What is the nature of mathematical reasoning? Is it deductive as commonly thought? Careful analysis shows us it is nothing of the kind; that it participates to some extent in the nature of inductive reasoning and for that reason is fruitful.

In other words mathematics is more akin to a science and it's a false economy to turn sciences into axiomatic mathematics.

Answer 5

As @Geremia pointed out, finding axioms for the mathematical portion of physics has certainly been proposed, and undoubtedly attempts have been made. However: Even if such axioms could be found, sufficient to cover all known phenomenae, and amenable as a basis for the mathematical portion of physics, there would still be the entirely separate issue of checking whether the world conforms to those axioms, and that is something no axioms can address.

Also, as @Jenhawk777 points out, the very validity of doing physics requires a lot of assumptions, which could be called 'axioms' - but I think that doesn't quite address the question asked (yes?).

Answer 6

To deconstruct and construct outcomes we need an approach that is systematic. Even if it is a random sample that we observe we need a systematic approach to evaluate it. The validity or the correctness of the scientific method and the axiomatic approach is a non-starter, this is because they are set to to constantly evolve. You can set the axioms and try to describe the outcome and if it doesn't then we keep on changing the axioms. If our hypothesis is wrong we keep on changing it in the scientific method. What we need is clarity, I.e. the limitations of our axioms and hypothesis. If you step outside of the frame "science" and label it as information(knowledge or outcomes) the purpose of scientific method and the axiomatic approach will be clear. To add to this, seperation of information into fields of study is also another deconstruction for simplicity and to evaluate information based whether if it belongs to the feild of mathematics or physics is s bad approach, for information is boundless...