Suppose I could somehow observe their inner mental activity directly. — RogueAI

If someone were traveling close to the speed of light relative to me... — RogueAI

Falsification was first developed by Karl Popper in the 1930s. Popper noticed that two types of statements are of particular value to scientists. The first are statements of observations, such as 'this is a white swan'. Logicians call these statements singular existential statements, since they assert the existence of some particular thing. They can be parsed in the form: there is an x which is a swan and is white.

The second type of statement of interest to scientists categorizes all instances of something, for example 'all swans are white'. Logicians call these statements universal. They are usually parsed in the form for all x, if x is a swan then x is white.

Scientific laws are commonly supposed to be of this form. Perhaps the most difficult question in the methodology of science is: how does one move from observations to laws? How can one validly infer a universal statement from any number of existential statements?

Inductivist methodology supposed that one can somehow move from a series of singular existential statements to a universal statement. That is, that one can move from ‘this is a white swan', “that is a white swan”, and so on, to a universal statement such as 'all swans are white'. This method is clearly logically invalid, since it is always possible that there may be a non-white swan that has somehow avoided observation. Yet some philosophers of science claim that science is based on such an inductive method.

Popper held that science could not be grounded on such an invalid inference. He proposed falsification as a solution to the problem of induction. Popper noticed that although a singular existential statement such as 'there is a white swan' cannot be used to affirm a universal statement, it can be used to show that one is false: the singular existential statement 'there is a black swan' serves to show that the universal statement 'all swans are white' is false, by modus tollens. 'There is a black swan' implies 'there is a non-white swan' which in turn implies 'there is something which is a swan and which is not white'.

Although the logic of naïve falsification is valid, it is rather limited. Popper drew attention to these limitations in The Logic of Scientific Discovery, in response to anticipated criticism from Duhem and Carnap. W. V. Quine is also well-known for his observation in his influential essay, "Two Dogmas of Empiricism" (which is reprinted in From a Logical Point of View), that nearly any statement can be made to fit with the data, so long as one makes the requisite "compensatory adjustments." In order to falsify a universal, one must find a true falsifying singular statement. But Popper pointed out that it is always possible to change the universal statement or the existential statement so that falsification does not occur. On hearing that a black swan has been observed in Australia, one might introduce ad hoc hypothesis, 'all swans are white except those found in Australia'; or one might adopt a skeptical attitude towards the observer, 'Australian ornithologists are incompetent'. As Popper put it, a decision is required on the part of the scientist to accept or reject the statements that go to make up a theory or that might falsify it. At some point, the weight of the ad hoc hypotheses and disregarded falsifying observations will become so great that it becomes unreasonable to support the theory any longer, and a decision will be made to reject it.

In place of naïve falsification, Popper envisioned science as evolving by the successive rejection of falsified theories,rather than falsified statements. Falsified theories are replaced by theories of greater explanatory power. Aristotelian mechanics explained observations of objects in everyday situations, but was falsified by Galileo’s experiments, and replaced by Newtonian mechanics. Newtonian mechanics extended the reach of the theory to the movement of the planets and the mechanics of gasses, but in its turn was falsified by the Michelson-Morley experiment and replaced by special relativity. At each stage, a new theory was accepted that had greater explanatory power, and as a result provided greater opportunity for its own falsification.

Naïve falsificationism is an unsuccessful attempt to proscribe a rationally unavoidable method for science. Falsificationism proper on the other hand is a prescription of a way in which scientists ought to behave as a matter of choice. Both can be seen as attempts to show that science has a special status because of the method that it employs. — Banno

• The claim only works if restricted to synthetic empirical universals in open domains.
• Shown to be true/false here means being empirically demonstrated.

The Sun is not what we see when we look at the sun. — AmadeusD

Neither set is countable — frank

The natural numbers are also a proper subset of the rationals, but they're the same size. — Srap Tasmaner

But Cantor showed that there is a way to force them through a chute so that you can count them one-at-a-time. It's interesting that it turns out you cannot do this with the real numbers. — Srap Tasmaner

OK. It depends on what you are doing. I was thinking of the point of origin on a graph, but that's not quite the same as counting numbers. — Ludwig V

I think many people believe that if something is referred to, it counts as an object. — Ludwig V

Numerals get their identity from roles in activities, not from reference to entities.
— Banno
You are not wrong. But now we are getting into trouble with the difference between numerals and numbers. I have a feeling, however, that we may need numbers in order to identify correspondences between numeral systems and perhaps even number systems with different bases. — Ludwig V

But don't we also need 0, as the starting-point? — Ludwig V

I don't have a problem about the timeless present in the case of constitutive norms. But in relation to procedures, I do. For the obvious reason, that a procedure takes place in time. — Ludwig V

You'll find that you can't. — frank

I've just been observing the different stances people are taking. — frank

Ok. I don't object to that. I doubt you can do the same thing for ZFC, though. So are you now suspicious that ZFC might be bullshit? — frank

An object in your mind is called a mental object. An object in your hand is a physical object. An abstract object is something that isn't physical, but it's not simply mental either. — frank

This is at the core of the problems with US politics. — Christoffer

• Benacerraf’s identification problem
• Gödel’s incompleteness theorems shows that set theory cannot fully capture all mathematical truths
• Category theory, topology, homotopy type theory don’t naturally live in the set-theoretic universe

There aren't a lot of experts in phil of math — frank

You have the minority view — frank

You're trying to have your cake and eat it too. — frank