L&FP, 67: So-called “critical rationalism” and the blunder of denying [defeat-able] warrant for knowledge

_{kairosfocus
March 25, 2023

Academic Freedom, Agitprop, knowledge, Logic and Reason

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IEP summarises:

“Critical Rationalism” is the name Karl Popper (1902-1994) gave to a modest and self-critical rationalism. He contrasted this view with “uncritical or comprehensive rationalism,” the received justificationist view that only what can be proved by reason and/or experience should be accepted. Popper argued that comprehensive rationalism cannot explain how proof is possible and that it leads to inconsistencies. Critical rationalism today is the project of extending Popper’s approach to all areas of thought and action. In each field the central task of critical rationalism is to replace allegedly justificatory methods with critical ones.

A common summary of this is that it replaces knowledge as justified, true belief, with “knowledge is unjustified untrue unbelief.” That is, we see here the ill advised privileging of hyperskepticism.

The quick answer is to update our understanding, based on how well informed people of common good sense generally use “knowledge.” Knowledge is a term of the people, not some abstruse, rarefied, dubious philosophical notion. And it is a term that is sound,

Namely, and following Plantinga, Gettier and others, knowledge is warranted, credibly true [and reliable] belief, i.e. it includes strong form cases where what is known is absolutely certain, AND a wider, weaker sense where what we claim to know is tested and found reliable, but is open to correction for cause. Newtonian dynamics counted as knowledge before the rise of modern physics and with modification to recognise limitations it still counts as knowledge. This is a paradigm case.

But doesn’t that come down to the same thing as critical rationalism and its focus on what is hard to criticise as what counts for now as “knowledge”?

Not at all.

First, the confident but open to correction spirit of warrant and tested reliability is utterly different from the cramped, distorted thought that naturally flows from the blunder of privileging selective or even global hyperskepticism.
Second, inference to the best explanation and wider observational, inductive approaches — the vast majority of common, day to day knowledge and professional practice — is not put under the chilling effect of dismissive, undue suspicion.
Third, knowledge is accepted as a commonplace phenomenon, not a privilege of the elite few, undermining the subtext of contempt that reeks out of far too much of skeptical discussion.

And if you imagine these considerations are of little weight, that is because you are part of the problem. END

Comments

Given what Popper says about the nature of philosophy and the nature of science in Conjectures and Refutations, I am skeptical that there is any easy version of argument that shows that Popper exempts himself from critical standards that he applies to others. His views are not self-undermining or self-defeating. It might very well be the case that Popper's views do not actually succeed by his own standards, but that's a different issue. @56
It is certainly consistent with how we experience the world.
Sure, but that tells us nothing about what justifies the assertion. All of us use the concepts we've acquired to understand the world as we experience it, and that holds true for "2+2=4" as much as it does for "most swans are white". Yet those assertions are justified in different ways, which is the whole point of insisting on a distinction between a priori and a posteriori here. Consider the alternative: suppose we give up on the a priori/a posteriori distinction. We could do this either by insisting that all knowledge is a priori and that everything based on experience is just opinion (if not colossal delusion). Or we could do that by insisting that all knowledge is a posteriori, and that even logic and mathematics could be revised in light of experience, just as "all swans are white" was revised in light of the discovery (to Europeans) of black swans. I hope I am not alone in finding both options unacceptable. Nevertheless, we do of course use the assertion "2+2=4" to do work in the world as we experience it, but that does not confirm the assertion, because it is an a priori truth -- it is logically entailed by the axioms of arithmetic. 2+2=4 would be equally justified in any world where no one had invented the practice of counting. To be sure, the axioms of arithmetic would probably not be discoverable in a world where there were no cognitive beings, or if the cognitive beings were mere spectators incapable of action, or if the actions they were capable of did not require them to distinguish between more than one thing of a certain kind, etc. We can certainly specify what a world must be like in order for the axioms of arithmetic to be discoverable by beings in that world. But that is categorically distinct from what justifies the axioms, since the axioms are said to hold in all possible worlds, regardless of whether they are discoverable by being in that world or not -- that's just what it means to say that they are necessary truths, not contingent truths.PyrrhoManiac1_{March 27, 2023
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For crying out loud, no one is trying to silence or marginalize your “corrections.” You could, however, be a little less pedantic about the whole thing…….
I've never seen KF post in an environment he couldn't control. He may have done and I've missed it but I doubt he would have received the courtesy to which he thinks himself entitled and demands here while dishing out ad homs to critics with gay abandon. Don't get me wrong, I don't complain. Every tub stands on its own bottom and any website can set and enforce its own standards of participation. At least token participation by some ID skeptics is currently tolerated here. I think this is the last pro-ID site I know of now. The crankiness and downtime make me wonder if it may not be for much longer. There's no new generation of ID-proponent youngsters ready to replace the old guard.Alan Fox_{March 27, 2023
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KF For crying out loud, no one is trying to silence or marginalize your "corrections." You could, however, be a little less pedantic about the whole thing.......chuckdarwin_{March 27, 2023
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CR @
CR: How did you come to choose that particular proposition as a candidate for immunity from criticism? Why 2+2=4 instead of, say, 3+4, the theorem of Pythagoras or all swans are white?
I have already answered this question. I will copy and paste my previous answer here:
Ori: Because 2+2=4 is conveniently confirmable by observing one’s fingers. In this context, 2+3=5 is a solid alternative.
PM1@
PM1: I would conjecture that Origenes likes this example because it illustrates the distinction between a priori and a posteriori assertions. Whatever justifies the assertion that 2+2=4, it is not going to be anything about how we experience the world.
It is certainly consistent with how we experience the world.Origenes_{March 27, 2023
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"Or half-a-bee. (Monty Python prompt if anyone cares.)." Eric. Andrewasauber_{March 27, 2023
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What we see here is a position on positions. A position that states that no position can be positively justified. Applied to itself, it undercuts itself and runs into humiliating self-contradiction. However, I am sure that we will be told that it cannot be applied to itself …. First, you mean like ID cannot be applied to ID's designer? Cleary, you understand the concept. Second, to quote Popper?
The question about the sources of our knowledge . . . has always been asked in the spirit of: ‘What are the best sources of our knowledge—the most reliable ones, those which will not lead us into error, and those to which we can and must turn, in case of doubt, as the last court of appeal?’ I propose to assume, instead, that no such ideal sources exist—no more than ideal rulers—and that all ‘sources’ are liable to lead us into error at times. And I propose to replace, therefore, the question of the sources of our knowledge by the entirely different question: ‘How can we hope to detect and eliminate error?’
Here, the term "propose" is explicitly used. Other times it is implied by, well, being a fallibilist. We first start with a problem, conjecture ideas of how to solve them, then try to find errors they contain. Not finding any errors does not positively justify them. They simply have held up better than other competing theories. As for the lack of justification...
Fallibilism, correctly understood, implies the possibility, not the impossibility, of knowledge, because the very concept of error, if taken seriously, implies that truth exists and can be found. The inherent limitation on human reason, that it can never find solid foundations for ideas, does not constitute any sort of limit on the creation of objective knowledge nor, therefore, on progress. The absence of foundation, whether infallible or probable, is no loss to anyone except tyrants and charlatans, because what the rest of us want from ideas is their content, not their provenance: If your disease has been cured by medical science, and you then become aware that science never proves anything but only disproves theories (and then only tentatively), you do not respond “oh dear, I’ll just have to die, then.”
Given that I've posted this quote multiple times, this is yet another example of wack-a-mole behavior. When one misconception is clarified, some other prior misconception is presented, etc. From another comment...
Of course this is no help for dogmatists who seek stronger reasons for belief, but that is a problem for them, not for exponents of critical preference.
IOW, as a justificationist, you're projecting your problem on me.critical rationalist_{March 27, 2023
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Jerry: other type of numbers are just mental abstracts and extremely useful for real world decisions. Arguably, when you append units to numbers then fractions become real values. Granted you can usually restate the value as a whole number under different units but that can get pretty silly. For example: if you lost half your arm in an accident there's not much point in defining a subunit of a arm so that you can refer to what you have left as so many whatevers when it's pretty clear what half-an-arm means. Or half-a-grapefruit. Or half-a-bee. (Monty Python prompt if anyone cares.). Also, some units (like moles) do not have the same physical (volume, pressure, etc) values from element to element so the idea of half-a-mole is pretty concrete once you specify the substance. Also I guess there's an argument for quarter-notes and eighth-notes in music. Not sure what the minimal unit of musical time would be . . . Don't think there is a sub-unit of temperature either, just degrees and fractions of degrees.JVL_{March 27, 2023
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I would conjecture that Origenes likes this example because it illustrates the distinction between a priori and a posteriori assertions. Whatever justifies the assertion that 2+2=4, it is not going to be anything about how we experience the world.
This does not conflict with Origenes having criticizing them by "[thinking] of ways or reasons they might have been conceivably false." IOW, Origenes would be a fine example of a practicing fallibilist. Another example? The original question... To paraphrase, "If fallibilism is true, then what of 2+2=4 or Pythagoras?" IOW, Origenes is criticizing fallibilism by proposing ways that it could be false. This is fallibilism at work. Just because we criticize an idea and the result is a lack of good criticisms, does this mean that criticism did not occur in the first place? No. It just means we currently lack good criticisms of that idea. Another example? Kant claimed that geometry is was a priori.
Kant associated geometric knowledge with the pure intuition of space; to know that an isosceles triangle (i.e. with two equal sides) has two equal angles at the base, the mathematician must produce a particular construction that makes the truth of the claim demonstrable,
At the time we had no good criticisms of Pythagoras theory. That is, until general relativity. In which case the angles may not always add up to exactly 180 degrees due to the warping of space. So, again, I'd suggest things like 2+2=4, Pythagoras theory, etc. do not reflect some special category. Rather they are ideas that, as of this very moment, we currently lack good criticism of. 2+2=4 is extremely hard to vary idea.critical rationalist_{March 27, 2023
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CR@48
According to the stance of critical preference no position can be positively justified but it is quite likely that one, (or some) will turn out to be better than others are in the light of critical discussion and tests.
What we see here is a position on positions. A position that states that no position can be positively justified. Applied to itself, it undercuts itself and runs into humiliating self-contradiction. However, I am sure that we will be told that it cannot be applied to itself ….
First, the stance of critical preference is not a position, it is a metacontext …
Aha. So, a position on positions is not a position, but a ‘metacontext’ instead. Does that mean that a metacontext, unlike a ‘position’, can be positively justified? Suppose an alternative metacontext on positions, one that says that some positions can be positively justified. What is the way forward here? Why would one accept the metacontext offered by critical rationalism over this alternative metacontext?Origenes_{March 27, 2023
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Whatever justifies the assertion that 2+2=4, it is not going to be anything about how we experience the world
it is a definition. The only numbers that exist in the real world are positive integers. We give a name to each. All been discussed before. As I said UD just goes in circles discussing the obvious. Or is it nonsense that is discussed. Hard to tell sometimes. Aside: other type of numbers are just mental abstracts and extremely useful for real world decisions. People often misinterpret usefulness for real.jerry_{March 27, 2023
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@46
How did you come to choose that particular proposition as a candidate for immunity from criticism? Why 2+2=4 instead of, say, 3+4, the theorem of Pythagoras or all swans are white? Was it because you decided that proposition would be the best to make your point because it was the most obvious, unambiguous truth of all you considered using?
I would conjecture that Origenes likes this example because it illustrates the distinction between a priori and a posteriori assertions. Whatever justifies the assertion that 2+2=4, it is not going to be anything about how we experience the world. I did find a nice passage on a priori knowledge in Conjectures and Refutations where he discusses Kant's criticism of Hume. It reminds me somewhat of Konrad Lorenz's physiological interpretation of the a priori. @47
But when directly confronted with the ominous threat of self-contradiction Popper changes his stance completely. On second thought, his cherished theory is, unlike all other theories, not falsifiable. His precious little theory is exempt from criticism after all:
I think that you are misreading Popper here. He is not saying that his account of science is exempt from criticism, only that is it exempt from the kind of criticism that is necessary for science, which is falsifiability. And it is exempt from that kind of criticism because his account of science is a philosophical account: he is not offering a scientific theory of what science is. To be entitled to this, he would need some distinction between philosophy and science. And indeed, this is precisely what he develops -- at least in Conjectures and Refutations, if not before. He suggests that the key difference between philosophical theories and scientific theories is that the former are irrefutable. This does not mean that they are exempt from criticism, but that they cannot be criticized in the same way that scientific theories are criticized. This raises the question, "by what criterion or standard shall a philosophical theory be criticized, if not empirical refutability?" And Popper's answer is: by how well the theory solves the problem that it was designed to solve, and whether there are better formulations of the problem, or solutions to it that have less unpalatable implications. For example, Kant's elaborate theory of the finite rational mind is designed to solve the problem of how to reconcile the determinism entailed by classical mechanics with the libertarianism presupposed by morality. We can ask if this problem should be taken seriously in light of 21st century science, and we can ask if Kant's solution to this problem is consistent with 21st century science (including cognitive science). I see no reason why we could not criticize Popper's theory by his own lights: what are the problems that his theory of science is intended to solve? Are those problems worth taking seriously? Are there solutions to those problems that make better sense to us? (If Popper himself would balk at being criticized in that way, so much the worse for him!)PyrrhoManiac1_{March 27, 2023
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From this essay on dogmatic 'true belief' framework of Western thought.
In the light of Bartley's ideas we can discern a number of possible attitudes towards positions, notably those of relativism, dogmatism (called “fideism” in the scholarly literature) and critical preference (or in Bartley's unfortunately clumsy language, “pancritical rationalism”.) Relativists tend to be disappointed dogmatists who realise that positive confirmation cannot be achieved. From this correct premise they proceed to the false conclusion that all positions are pretty much the same and none can really claim to be better than any other. There is no such thing as the truth, no way to get nearer to the truth and there is no such thing as a rational position. Fideists are people who believe that knowledge is based on an act of faith. Consequently they embrace whatever they want to regard as the truth. If they stop to think about it they may accept that there is no logical way to establish a positive justification for their beliefs or any others, so they insist that we make our choice regardless of reason: ”Here I stand!”. Most forms of rationalism up to date have, at rock bottom, shared this attitude with the irrationalists and other fundamentalists because they share the same 'true belief' structure of thought. According to the stance of critical preference no position can be positively justified but it is quite likely that one, (or some) will turn out to be better than others are in the light of critical discussion and tests. This type of rationality holds all its positions and propositions open to criticism and a standard objection to this stance is that it is empty; just holding our positions open to criticism provides no guidance as to what position we should adopt in any particular situation. This criticism misses its mark for two reasons. First, the stance of critical preference is not a position, it is a metacontext and as such it is not directed at solving the kind of problems that are solved by adopting a position on some issue or other. It is concerned with the way that such positions are adopted, criticised, defended and relinquished. Second, Bartley does provide guidance on adopting positions; we may adopt the position that to this moment has stood up to criticism most effectively. Of course this is no help for dogmatists who seek stronger reasons for belief, but that is a problem for them, not for exponents of critical preference.
I'm not a disappointed dogmatist, KF would like to paint me as. Of course, I've referenced and even quoted this several times before. Yet he continues to present the same false dilemma. What gives?critical rationalist_{March 27, 2023
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~Critical Rationalism is itself Beyond Criticism~ Popper on “all knowledge”:
“all knowledge is hypothetical” [OKN 30]; alternatively, “All knowledge remains... conjectural” [RASC xxxv]. In other passages it is “all theories” which are conjectural [eg OKN 80].
Is the ‘knowledge’ (or ‘theory’) that ‘all knowledge remains conjectural’ itself merely conjecture? IOW is Popper’s claim nothing but a self-contradictory blunder? Does the claim, applied to itself, undercut itself? Can we all just laugh about it (“what an idiot”) and move on? Popper himself seems to suggest that this is indeed the case when he writes:
“nothing is exempt from criticism... not even this principle of the critical method itself” [OSE2 379].
But when directly confronted with the ominous threat of self-contradiction Popper changes his stance completely. On second thought, his cherished theory is, unlike all other theories, not falsifiable. His precious little theory is exempt from criticism after all:
Dykes: In “Replies to my Critics” Popper attempted to deflect this line of questioning by resorting to his demarcation criterion: “my theory is not empirical, but methodological or philosophical, and it need not therefore be falsifiable. Falsifiability is a criterion of demarcation, not one of meaning” [PKP2 1010]. Eh? Admittedly, thirty years separate the two quotations, but defending fallibilism by implying that philosophy is not empirical? Dear me. Where is that going to take us? I mean, where did Popper get his facts about philosophy from? Besides, only a hundred or so pages later, he came right out and said he was an empiricist [PKP2 1121].34
So, e.g. Kuhn’s model of the growth of knowledge can be criticized and rejected, but that does not go for Popper’s critical rationalism. Here I share Dykes’s indignation. When cornered the true childish intent of hyper-skepticism reveals itself: all your knowledge is garbage, but mine is not. Your knowledge will be criticized forever and ever, but not mine.Origenes_{March 27, 2023
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I note here that the theory ‘all swans are white’ differs in kind from e.g. the theory that 2+2=4.
I’d note that you have for at least the third time picked 2+2=4 as a shining example of an absolute, axiomatic truth, that is immune from criticism. How did you come to choose that particular proposition as a candidate for immunity from criticism? Why 2+2=4 instead of, say, 3+4, the theorem of Pythagoras or all swans are white? Was it because you decided that proposition would be the best to make your point because it was the most obvious, unambiguous truth of all you considered using? How did you determine how obvious and unambiguous each of those candidate propositions were, compared to the others? Did you not think of ways or reasons they might have been conceivably false? Did you not, well, criticize them?critical rationalist_{March 27, 2023
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FP & CD, I suggest, you find a way to make a substantial contribution. Perhaps, you are unaware that the errors, fallacies and sometimes agit prop techniques we have corrected and pointed out substantially here at UD are in fact major and all too common problems in our day and age. The pointing out of these problematic approaches here at UD was not made up from whole cloth, the fallacies were identified on specific substance, for cause. And if you imagine that pretending that such concerns are groundless you can silence or marginalise their correction [so, letting such misbehaviour take over discussion and frustrate sound progress], you are sadly mistaken. Indeed beyond a point you will simply confirm to one and all that you are playing the irresponsible, destructive troll. KFkairosfocus_{March 26, 2023
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PM1, actually, structure and quantity has aspects that are part of the being of any world. The unique null set is relevant to any world, and the set that collects it etc will be just as unique, i.e. N is manifest in any world as part of its framework. From N we have ZQRCR* also, with all that such entails. These are part of the logic of being and shape possibilities by necessary relations including say 2 + 3 = 5, and of course the first Pythagorean triplet, 345, etc. As one consequence, when Young identified wave interference, an objection made was then the shadow of a small sphere should have a dot of light in its shadow's centre, which is silly. But someone checked, and the dot was indeed there, just tiny and easily overlooked. Similarly, at Copenhagen 1926, Einstein made an elaborate objection to uncertainty, only to end up inadvertently discovering a second major form, energy-time. And there are many other cases where logic of being expressed through structure and quantity has shown how it is framework to reality. KFkairosfocus_{March 26, 2023
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PM1/42 Excuse my presumption. I stand corrected.......chuckdarwin_{March 26, 2023
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@39
No one in the skeptic camp “despises” you or anyone else affiliated with ID, nor have I seen any remarks from skeptics and other non-IDers that would so suggest……
Eh, speak for yourself! I certainly do despise many people who also support some version of intelligent design. But I do not despise them because they support intelligent design. I despise them on other grounds; the fact that they also suppose ID has almost nothing to do with it.PyrrhoManiac1_{March 26, 2023
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@31 Agreed that the axiomization of a domain comes at the culmination of its development, but for those of us who distinguish between the logic of discovery and the logic of justification, this is not a huge problem. To prove that 2+2=4, we need to define some primitive notions, such as the concept of a set, membership in a set, succession, and identity. If we can define those, we can define the empty set as {}. And we can include a set that has the empty set as a member: {{}}. And since we can repeat that operation infinitely many times, we can easily generate all the integers, and some operations performed on the integers give us all the reals. With enough built into the definitions, it becomes possible to prove that
{{}}, {{}} is identical to {{{{}}}}
which becomes 2+2=4 with a use of a convenient notation. And while this is provably irreducible to first-order logic, thanks to Goedel, it has nothing to do with metaphysics or science, either. To say that mathematics concerns what must happen in any possible world (or a class of possible worlds) is to add a metaphysical gloss to a set-theoretic result. We can say that mathematics studies different kinds of formal structures, and leave it as a distinct issue exactly how to settle the metaphysics (and epistemology) of mathematics. @33
PM1, we are both aware that Popper and his falsificationism, tied to critical rationalism, are widely popularised views on science.
I can accept that it's an interesting question as to why Popper is the most popular philosopher of science amongst practicing scientists. And I think there's a straightforward answer: because Popper endorses a romanticized conception of the scientist as exemplifying critical thought, while at the same time thinking of scientific theories as solutions to interesting puzzles rather than as revealing how things really are in some metaphysically deep or important sense. The scientist as extremely sophisticated problem-solver is important to how most scientists think about themselves, and that's what Popper gives them.
Where, with scientism, many take that to cover all knowledge.
Maybe. I have some misgivings about the term "scientism", which I've gone into exhaustive depth elsewhere on this blog. For now, I think it suffices to note that, so far as I can tell, Popper would not say that science is the only kind of knowledge -- he could not say that and also have views about what makes science distinct and different from other kinds of knowledge. One thing about the SEP entry that intrigued me is partly biographical about Popper. He was interested in Marxism and in psychoanalysis when he was younger. Then he encountered Einstein. And the main difference is that Einstein wanted to see if he was wrong: he wanted the physics community to do experiments which would disconfirm general relativity. By contrast, Marxists and Freudians claimed to be scientific but had no interest in having their pet theories disconfirmed -- on the contrary, any evidence that went against their theories was just dismissed. I think Popper is onto something very important about the attitude that exemplifies science at its best, and that might be quite different from how everyday empirical knowledge works. I doubt Popper would object to someone who says, "I know that Bob is going to ask me for money today, because we get paid tomorrow and Bob always asks me for money right before we get paid, because he always spends his money faster than he earns it." That's a reliable generalization from past experience, and as inductively valid as everyday empirical knowledge gets. Yet it doesn't count as scientific knowledge, and we can think of several reasons why it doesn't. For one thing, it doesn't yield any insight into laws of nature (if there are any) and it doesn't unify lots of different phenomena under a more comprehensive principle. It doesn't elucidate some underlying mechanism that explains why Bob spends his money faster than he earns it. But more importantly, I think for Popper, all it really amounts to is a subjective estimate of what's likely to happen -- and that's why it fails to count as genuinely objective knowledge. Importantly, it should be added, Popper was not a physicalist or materialist. He thought the world of cultural objects (including works of art and scientific theories) was irreducible to the world of mental states or the world of physical objects (and also that the world of mental states is irreducible to the world of physical objects).PyrrhoManiac1_{March 26, 2023
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FP @38
Thank you for adding to the list of disingenuous tactics used by some rather than substantively addressing their arguments.
You have posted no less than 3 times in this thread. What exactly is the substantive argument that you are making? What did you offer that can be substantively addressed?Origenes_{March 26, 2023
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KF No one in the skeptic camp "despises" you or anyone else affiliated with ID, nor have I seen any remarks from skeptics and other non-IDers that would so suggest......chuckdarwin_{March 26, 2023
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Kairosfocus writes:
we cannot but note your choice of distractors, which manifest the precise pattern of projection to the despised other. And, swarm tactics are a notorious mob pattern….That sidestepping is a clear sign that you have little of substance but wish to poison [the well].
Thank you for adding to the list of disingenuous tactics used by some rather than substantively addressing their arguments. I knew that I had forgotten some.Ford Prefect_{March 26, 2023
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FP, we cannot but note your choice of distractors, which manifest the precise pattern of projection to the despised other. And, swarm tactics are a notorious mob pattern. You would have been well advised, instead, to address substantial matters, or else, perhaps to monitor and learn. That sidestepping is a clear sign that you have little of substance but wish to poison. Consider yourself further exposed for record. KFkairosfocus_{March 26, 2023
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CD writes:
And don’t forget everyone’s pop-psych favorite, “ cognitive dissonance.”
Very true. But we also can’t forget the accusation of “piling on” whenever two or more ID opponents point out the easily confirmed disingenuous tactics used by some of the UD parishioners.Ford Prefect_{March 26, 2023
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PS, to see where this goes, A slice of Hilbert's axiomatisation:
I. Incidence For every two points A and B there exists a line a that contains them both. We write AB = a or BA = a. Instead of “contains,” we may also employ other forms of expression; for example, we may say “A lies upon a”, “A is a point of a”, “a goes through A and through B”, “a joins A to B”, etc. If A lies upon a and at the same time upon another line b, we make use also of the expression: “The lines a and b have the point A in common,” etc. For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ? C, then also BC = a. There exist at least two points on a line. There exist at least three points that do not lie on a line. For every three points A, B, C not situated on the same line there exists a plane ? that contains all of them. For every plane there exists a point which lies on it. We write ABC = ?. We employ also the expressions: “A, B, C, lie in ?”; “A, B, C are points of ?”, etc. For every three points A, B, C which do not lie in the same line, there exists no more than one plane that contains them all. If two points A, B of a line a lie in a plane ?, then every point of a lies in ?. In this case we say: “The line a lies in the plane ?,” etc. If two planes ?, ? have a point A in common, then they have at least a second point B in common. There exist at least four points not lying in a plane.
More refined but not substantially different in spirit. KFkairosfocus_{March 26, 2023
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PM1, to flesh out my point, here is a summary of Euclid's framework, from Wikipedia:
Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):[5] Let the following be postulated: 1 To draw a straight line from any point to any point. 2 To produce (extend) a finite straight line continuously in a straight line. 3 To describe a circle with any centre and distance (radius). 4 That all right angles are equal to one another. 5 [The parallel postulate]: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. Although Euclid explicitly only asserts the existence of the constructed objects, in his reasoning he also implicitly assumes them to be unique. The Elements also include the following five "common notions": 1 Things that are equal to the same thing are also equal to one another (the transitive property of a Euclidean relation). 2 If equals are added to equals, then the wholes are equal (Addition property of equality). 3 If equals are subtracted from equals, then the differences are equal (subtraction property of equality). 4 Things that coincide with one another are equal to one another (reflexive property). 5 The whole is greater than the part.
We can no longer assume familiarity.kairosfocus_{March 26, 2023
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PM1, we are both aware that Popper and his falsificationism, tied to critical rationalism, are widely popularised views on science. Where, with scientism, many take that to cover all knowledge. So, especially given a commenter using such tactics, it is highly relevant to address some of his thought. And Origenes is right to ask some pretty pointed questions. KFkairosfocus_{March 26, 2023
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PM1 @27
PM1: Popper is very clearly distinguishing between scientific theories and philosophical accounts of scientific theories.

Ori: Can you quote Popper making this distinction?

PM1: No
Can you explain what makes you say that "Popper is very clearly distinguishing between scientific theories and philosophical accounts of scientific theories"?Origenes_{March 26, 2023
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PM1, Mathematics is not simply an arbitrary game and logic is likewise. Before we get to axiomatisations and systems [esp those of C19 - 20] first we have built up a fund of mathematical facts of high reliability, such as || + ||| --> |||||, this is not just a game where glyphs are chained, 2 + 3 = 5. Likewise 3^2 + 4^2 = 5^2, with a right angle triangle enfolded at the points of tying the squares together is a fact embedded in the twelve segment rope, long before the general theorems were worked out. Axiomatisations became accepted as they were tying principles that unified the acknowledged facts, extended structures and enabled proving further useful results, in the first instance, then we developed the idea of general axiomatisations [including of algebra-icised logic]. Then, at length Godel incompleteness struck. So, we find out that truth and knowledge bridge the Kantian ugly gulch. I suggest, core Mathematics is an aspect of logic of being addressing possible worlds, dealing with structure and quantity, e.g. starting with {} --> 0 and articulating the von Neumann construction etc. Axiomatisations then articulate hopefully possible logic model worlds and key necessary being entities and associated necessary states of affairs of structure and quantity extend to any possible world. Hence, an answer to Wigner's wonder. KFkairosfocus_{March 26, 2023
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Do we really know in what kind of world we live? An idiot in a comment made comparisons between a child and a pig. https://rumble.com/v2b1ujq-eyes-of-the-devil-a-documentary-film.htmlwhistler_{March 26, 2023
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