I came across the notion of falsifiability quite recently.
The wikipedia article on the same states that:

Falsifiability or refutability of a statement, hypothesis, or theory is the inherent possibility that it can be proven false. A statement is called falsifiable if it is possible to conceive of an observation or an argument which negates the statement in question. In this sense, falsify is synonymous with nullify, meaning to invalidate or "show to be false".
For a statement to be questioned using observation, it needs to be at least theoretically possible that it can come into conflict with observation.

While I can understand the general concept - I would like to have a deeper understanding of the same. Popper mentions that this notion differentiates science from pseudo - science.

Can someone please give me some examples for the same? - So that I might understand the idea more intuitively. Specifically if you could provide what would be the falsifiability arguments/observations would be for:

  • Newton's theory of gravitation.
  • Heliocentralism
  • Theorem of calculus.
  • Probability theory.

Basically two popular theories from the realm of physics and two popular theories from mathematics (which I might possibly be familiar with), would do. Need not be just these four.

  • Newton's theory: a free apple "falling" from the floor to the ceiling. Jul 2, 2016 at 15:29
  • Right - if we observe that theory of gravitation would be falsified.
    – user12196
    Jul 2, 2016 at 15:43
  • Of course, the more complex is the theory, more difficult is to found "simple" falsifying conditions like that. When many "factors" are involved, a falsifying experiment must "manage" all of them. Consider the well-known discovery of Neptune by Urbain Le Verrier : a potential "falsifier" has been transformed into a brilliant "verification". Jul 2, 2016 at 15:56
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    For mathematical theories, it is not so clear if Popper's criteria applies. In principle, we can say that the only way to "falsify" a math theory is proving his inconsistency. Jul 2, 2016 at 15:57
  • @MauroALLEGRANZA Can you give me some sources to study up Urbain Le Verrier's discovery - specifically how it relates to falsifiability.
    – user12196
    Jul 2, 2016 at 17:25

4 Answers 4


The best way to understand Popper is to read Popper. There are a few commentators who have correctly understood his ideas, but the vast bulk of commentary on Popper is not even able to state his ideas correctly. Lakatos, Feyerabend and Kuhn are especially bad and should be avoided.

To understand falsification properly, you need to understand Popper's theory of knowledge more broadly. Most philosophers of science who take science seriously and think it is good are inductivists: they believe in a process called induction. Induction supposedly involves (1) taking observations, (2) using them to make theories, and then (3) showing those theories are true or probably true by more observations. People have looked at many phenomena such as the night sky, biology, medicine and so on, without learning much for thousands of years. So just observing stuff doesn't do much good. If you don't know what to look for, just observing will not produce progress, so step (1) is impossible. In addition, explanations don't follow from observations. The theory of stars has implications for many events we will never observe, e.g. - supernovae that took place before there were human observers, and those events don't follow from observations without a theory of how stars change. So steps (2) and (3) are also impossible.

So if we don't get theories from observation how do we get them? We guess. You look for a problem: some issue that is not explained by current ideas. You guess solutions to that problem. You then criticise the proposed solutions. This criticism may involve experiments, but many theories can be eliminated without doing experiments, e.g. - inconsistent theories.

An experiment involves looking for a situation in which two or more different ideas about how the world works make different predictions. You then either set up that situation or look for an existing system that realises that situation. Newton's theory of gravity and Einstein's general theory of relativity made different predictions about Mercury, and Newton's theory was refuted.

Some philosophers make a lot of fuss about the possibility that you might do an experiment wrong or misinterpret the results. But as Popper pointed out in Logic of Scientific Discovery, Chapter V (especially Section 29), this problem is solved by his epistemology. If an experiment contradicts an existing theory, that's a problem. This problem can be solved by any guess that explains the difference and is not eliminated by some criticism. The discovery of Neptune was taken as an example above, so let's look at it. An unsolved problem was found in explaining the orbits of some planets. Urbain Le Verrier guessed that there might be another planet. He worked out some constraints on where the planet could be to produce such effects, Johann Gottfried Galle looked for it and found it. If Galle had not found the planet that problem would have remained unsolved. Perhaps some other explanation could have been found to reconcile Newtonian mechanics with observation, perhaps not. Popper recommended that a proposed solution to a scientific problem should be rejected if it was ad hoc: if it had no implications beyond the problem it was invented to solve.

I'm going to skip the heliocentric theory because it is fairly similar to Newtonian mechanics. If you want a long list of examples, see the introduction to "Realism and the Aim of Science" by Popper.

Mathematical theories are about abstractions. They can be critically discussed, but not experimentally tested. 1+1 = 2 even though it is possible to think of examples of putting two objects together and only getting one object as a result. If you move two piles of sand together, you may only get one pile. So you have to think carefully about what systems you take as models of mathematical operations such as addition. For a discussion see "Realism and the Aim of Science" by Popper Chapter III, Section 24.

As far as probability is concerned, the best existing explanations have been provided by David Deutsch, see


For explanations of Popper's positions, see "Objective Knowledge" by Popper, Chapter 1, "Realism and the Aim of Science" by Popper, "Logic of Scientific Discovery" by Popper, "The Fabric of Reality" by David Deutsch, Chapters 3 and 7, and "The Beginning of Infinity" by David Deutsch, Chapters 1,2,4 and 13.


In the comments to Cort Ammon's answer you say:

"So we can't falsify mathematical theories? I thought Popper's method was a way of distinguishing the scientific from the non scientific - does that imply mathematical constructs are not scientific or is there something wrong with Popper's method."

Exactly, mathematical theories are not scientific theories. Mathematics is about abstract mathematical objects, Science is about empirically observables phenomena. The truth of mathematical statements are prove using logic and reason alone, while the truth of statements in physics, chemistry, biology, etc...are proven by experiment and observation. This was best described by David Hume, with his distinction known as Hume's Fork:

"All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic ... [which are] discoverable by the mere operation of thought ... Matters of fact, which are the second object of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing." - An Enquiry Concerning Human Understanding

So things like the fundamental theorem of calculus and probability theory can't be falsified because they don't correspond to anything observable. They, like all mathematical truths are proved solely using the rules and axioms of logic.

This is the whole point of falsification, one has to attempt to show that they empirically observe a phenomena that contradicts their theory. So the Newton's theory of gravity says that apples should fall every time we let go of them in midair. Pre Popper's falsificationism, Newton's theory is falsified if someone raises an apple lets go of it, and instead of it falling it hovers in the air or goes upwards.

Similarly per Popper, heliocentrism will be falsified the day that Venus or Mars, or one of the other planets is observed in a different orbit then the one predicted by the theory.

This points to an interesting problem with Popper's theory, that of auxiliary hypotheses (also called the Duhem-Quine thesis, or they idea that all observations are theory laden): Consider that at the beginning of the 19th century the orbit of Uranus was different than what was predicted by Newtoninan mechanics and heliocentrism. But astronomers, instead of abandoning the theory, concluded that there was an unknown planet modifying the orbit of Uranus, which they later confirmed and called Neptune. So the dilemma is: When observation contradicts theory, is the theory falsified? or is there missing data that can explain the mismatch between theory and predictions?

The issue of how to solve the problem of auxiallry hypotheses is still debated, and hasn't been solved yet. See the ideas of W.V.O Quine, Thomas Kuhn, Imre Lakatos and Paul Feyerabend, all in response to Popper's concept of falsification.

  • The thing is that the wiki article also says arguments could be used to falsify-so I thought it might be applicable for mathematics constructs as well.
    – user12196
    Jul 3, 2016 at 19:01

Falsification is an excellent and easy to understand system in principle, but much more nuanced in implementation. The easiest to falsify hypotheses are those famous ones such as "all swans are white," which can be falsified by observing a black swan. Of course, this assumes we all agree on what is a swan is. Hypotheses get murkier from there.

When it comes to real meaningful scientific hypotheses, falsification is typically more of an extended process rather than an instantaneous event. A scientific theory which is falsifiable is one where some results could cast substantial doubt on the hypothesis, and that doubt can be compounded by future tests.

For example, if one believed the hypothesis that light acts as a wave, one would be surprised to see particle like behavior. The photoelectric effect is one such effect that we now know exhibits particle like behavior. The first time one observes particle like behavior from an experiment, one might assume the results were a measurement error. Doing it a second time would begin casting doubt on the theory that light always behaves like a wave. Having dozens of scientists all run such experiments multiple times, and each discovering particle like effects would eventually "falsify" the hypothesis.

This process is even more complicated due to statistics. If I claim there is a gaussian error term on my results, you can never truly prove that my theory is wrong, because there is always a non-zero chance that you simply observed random luck. However, in practice, once the probability of such chance events is low enough, we declare a theory "falsified." How high one has to go is discipline dependent. In sociology, we regularly see error terms permitting 10% or even 20% due to unexplained factors. In particle physics, a hypothesis is not declared "confirmed" until those unexplained factors account for no more than 0.00001% of the total observed effects. This is because subatomic particles behave quite regularly, and we're able to generate as many results as needed to attain such high degrees of confidence. In sociology, it is much harder to repeat experiments and there is a great deal of variance between individuals, so the best we can do is lower degrees of confidence.

As for your list of examples:

Newton's theory of gravitation.

It is generally accepted that the motion of planets is almost completely governed by gravitational interactions. If we were to observe the motion of the planets, and find substantial deviations not explained by his theory of gravity, this would either falsify his theory or show that there are other forces at work. I believe we actually do see results which would falsify his work: you have to account for relativity to explain some movements (particularly in cases near a black hole)


Heliocentralism actually cannot be proven nor disproven because it is merely a model. Its more akin to a coordinate system transform than a theory. However, if one assumes geocentralism, one is forced to admit many strange forces which account for all of the movement we see in the planets. If one assumes heliocentralism, the movement can be explained entirely with simple conservative gravity models like Newton's theory of gravity. It is the simplicity of the heliocentric model that made it so effective.

Theorem of calculus.

Consider if we couldn't go anywhere, because of Xeno's paradox. This would demonstrate that the assumptions we make regarding limits are false. That being said, calculus is a mathematical construct. All that we can really falsify is its usefulness in describing the world around us.

Probability theory.

Once again this is a mathematical construct, making it difficult to falsify. However, one could argue that it is "falsified" by demonstrating that it does not effectively model reality. One major assumption in much of probability is IID: the idea that observations are (I)ndependent (I)denticaly (D)istributed. If there was a reason to argue that this assumption is invalid in the real world, then much of probability would not apply. This actually does occur when exploring the human mind. In many cases the assumption of IID is very poorly founded, so many simplifications that probability would permit are simply invalid when discussing the behavior of the mind.

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    So we can't falsify mathematical theories? I thought Popper's method was a way of distinguishing the scientific from the non scientific - does that imply mathematical constructs are not scientific or is there something wrong with Popper's method. Theories in physics use many mathematical constructs as well - so if they are not falsifiable how come the concepts on physics are?
    – user12196
    Jul 3, 2016 at 8:33
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    @novice I cannot answer the question "does that imply mathematical constructs are not scientific" directly, because it would involve a detailed discussion of exactly what "not scientific" means to you. That would be better suited for a chat room, rather than comments. However, I will try my best to answer obliquely. The validity of a mathematical construct is based in the validity of its assumptions and the validity of the rules of inference associated with it, not physical reality. It is only as one seeks to apply said constructs to the real world that the concept of falsification...
    – Cort Ammon
    Jul 3, 2016 at 16:39
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    ... becomes meaningful. Without that application to physical reality, mathematical theories are subject to far more stringent validity requirements than falsification would ever ask from them. However, when applying such constructs to science, one does make the assumption that those constructs are indeed valid. If the real numbers do not, indeed, form a field, calculus falls apart. On the other hand, it is totally valid to have mathematical constructs which do not have an immediately obvious connection to reality. The concept of complex numbers, for instance, was a mathematical...
    – Cort Ammon
    Jul 3, 2016 at 16:41
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    ... curiosity until Euler's function connected them to cyclic motion. Now they are used constantly in science. If you are interested in this question of the validity of mathematical construct, I recommend a beautiful video by Vsauce, How to Count past Infinity. He does an excellent job of explaining the subtle distinction between scientific theories and mathematical ones.
    – Cort Ammon
    Jul 3, 2016 at 16:44
  • Thanks for your comments as well as the resource you have provided. I confess I am not sure how Popper gives arguments/justifications in favor of the notion of falsifiability to distinguish science from pseudo-science. I read the part about observation and argument - and thought that an argument for the falsifiability of mathematical constructs must be possible - instead of an observation. Let me go into this more deeply and come back with more questions.
    – user12196
    Jul 3, 2016 at 17:20

I'm not sure just how useful falsification is in explaining the actual progress of science, in the sense of the revision of its basic concepts; its not for example a source of new ideas, but of pruning out what is given on the basis of what is known. Its a minor mode of progress, and not its major mode. A major mode would tell us how to find new ideas, unfortunately such a philosphical stone is illusionary.

For example, calculus can be explained by attempting to give meaning to 0/0; this, in terms of the usual arithmetic operations, is nonsensical; however mathematicians like to 'close up' operations; 0/0 can in fact be given meaning by thinking of it as dx/dy; of course this opens up the whole new world of calculus.

Similarly, no meaning could be given to the square root of -1; eventually one was found that was useful: i -the imaginary; and it again opened up a whole new world of complex geometry.

Probability is a concept with intuitive appeal; yet Quantum Mechanics relies on the notion of the square root of probability, and in fact a great deal of the bizarre beahviour can be explained on the basis of this new concept which still hasn't found a properly ontological basis in the same way that the infinitesimal or the imaginary has.

  • -1 This answer does not even attempt to address the question.
    – MmmHmm
    Jun 23, 2018 at 15:15

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