How do you address my comment about subatomic particles, are you implying scientists strictly control subatomic particles? — leo

Can you link to any particular publication in order to clarify what it is about?

I don't agree there is such a thing as "the scientific method". — leo

The scientific method is an empirical method of acquiring knowledge that has characterized the development of science since at least the 17th century. It involves careful observation, applying rigorous skepticism about what is observed, given that cognitive assumptions can distort how one interprets the observation. It involves formulating hypotheses, via induction, based on such observations; experimental and measurement-based testing of deductions drawn from the hypotheses; and refinement (or elimination) of the hypotheses based on the experimental findings. These are principles of the scientific method, as distinguished from a definitive series of steps applicable to all scientific enterprises.

Knowledge, as a justified belief, is justified with standard epistemic methods, whereunder the scientific one.

Whatever method you have in mind, there are plenty of examples of scientists who didn't follow that method when they built their theory (yet their theories are considered to be 'scientific'), or there are plenty of examples of theories/practices that follow that method and yet are considered to be 'unscientific'. — leo

Can you give concrete examples for your view?

I mentioned above how that idea is flawed. Any observation has to be interpreted in order to say whether it is evidence of something. — leo

Arbitrary observations cannot be used for the purpose of validating scientific theories. It is not possible to establish causality between input and output without strictly controlling input. Furthermore, other researchers must be able to repeat the experimental tests in order to verify the claim. That is why only observations in a laboratory setting may be used in such experimental test reports.

Your views are far outside what is supported by the scientific method.

Isn't God supposed to resolve issues with the least violence, the most intelligence and compassion? — BrianW

You may be confusing peace with pacifism.

Pacifism does not lead to peace. Pacifism only leads to contempt. Peace can only exist in mutual respect, and all respect is ultimately always based on the fear for reprisals.

It often takes a hell of a lot of reprisals to finally bring peace.

One of the author's points is that QM as it stands does not prove that reality is random at a micro level - that's down to the interpretation of QM. — Devans99

I certainly agree with what Hrvoje Nikolic writes on the matter:

Fundamental randomness as a myth

Of course, if the usual form of QM is really the ultimate truth, then it is true that nature is fundamentally random. But who says that the usual form of QM really is the ultimate truth? (A serious
scientist will never claim that for any current theory.)

Visual observation requires receiving light on a phenomenon. This becomes a fundamental problem when the phenomenon observed is itself of the same size or smaller than light particles. In that case, we cannot expect an ordinary observation experience to occur. It will necessarily be confused.

Still, I am interested more on the Philosophy of Science, I want to understand more the activity we call Science, why hypothesis like God's existence are not considered scientific. I do not feel confident to read Popper's The Logic of Scientific Discovery, but, after all, what is actually science? — Jorge

Science is every proposition that can be justified by experimental testing.

Science is an epistemic domain, i.e. all knowledge that can be reached and justified with a particular epistemic method. God's existence is considered not a scientific question, in the sense that the scientific method cannot reach it in order to justify an answer.

Every epistemic method generates its own epistemic domain that does not intersect with other epistemic domains: mathematics, science, history, epistemology itself, and possibly other methods.

It is the same question as: Does true randomness exist?

At a macro-level, randomness is deemed not to exist:

What we normally call “random” is not truly random, but only appears so. The randomness is a reflection of our ignorance about the thing being observed, rather than something inherent to it.

According to the Bell experiments, the axiom of realism mostly falls apart -- but with loopholes -- at the scale of photons and electrons. The smaller the scale, the less the axiom of realism is sustainable.

Axiom of realism: The moon exists, even if we do not look at it.

If I understand it right, according to Bell's theorem, the axiom of causality requires the axiom of realism, the existence of which really depends on just the presence of loopholes that explicitly allow for local reality.

Generalized realism, however, seems to be unsustainable.

Yet, mathematical physics is one of the most successful sciences. Your theory can't explain this success. On it, what mathematical physicists do is completely unjustifiable. — Dfpolis

Mathematical physics is still physics. It is not axiomatic. It will ultimately still be experimentally tested. The amount of mathematics used by physics does not change its fundamental nature. It certainly does not turn physics into mathematics. It just makes sure that it is incredibly consistent. It is its consistency that explains its success.

The difference between physics and mathematics is not that one is about nature and the other not — Dfpolis

That is exactly the difference.

Math is about nature as quantifiable — Dfpolis

Mathematics is not number theory. Most mathematical theorems are not about numbers or quantities.

The reason for Russell's paradox is not some formal problem that requires a theory of types (though a theory of types avoids the problem). The reason for it is that there is nothing in reality from which we can abstract the concept of the set of all sets that do not include themselves, just as there is nothing in reality from which we can abstract the parallel postulate or the axiom of choice. — Dfpolis

You can represent a set by its membership functions and disregard what elements it contains. From there on, the paradox becomes a problem with these membership functions. The function will not manage to return a result, simply because it never stops running. That is how the problem appears when it is modeled in software. The way automated systems behave, is unrelated to physical-world problems that they would mirror, because they often don't, and in this case, they certainly don't.

All you're doing is ruling out obvious nonsense, leaving open the possibility that all mathematics may be obscure nonsense, — Dfpolis

Only category theory is termed general abstract nonsense.

In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory and homological algebra. More generally, “abstract nonsense” may refer to a proof that relies on category-theoretic methods, or even to the study of category theory itself.

Not all mathematics is abstract nonsense, but the very best stuff certainly is.

Now not all feminism is wrong ... — Ilya B Shambat

I think it is.

Women are more naturally beautiful than men. — Ilya B Shambat

Beauty certainly signals health, even though it may signal other things too. In a more primitive, more natural society, the ability to bear children (for women) and the ability to provide resources (for men) would have required both genders to be in good health. Both women and men had to be beautiful.

With development of technology, which started accelerating after humanity started farming, the gap may have started growing too. Merely handsome men may no longer, necessarily be good resource providers. Even though women may undoubtedly have a preference for naturally handsome men, they (and/or their families) may not always want them as husbands.

Ugly men could still have lots of offspring and be genetically successful. Ugly women ... not so sure about that. It is probably harder for them.

Epistemology looks at what knowledge is, what justification is, etc. Looking at knowledge and/or justificatio — Terrapin Station

Well, then we can ask ourselves the question: Is knowledge about knowledge, i.e. the metaknowledge, itself knowledge? If it is itself justified, then yes. Otherwise, no.

I think that epistemology is a justified belief, and therefore, itself knowledge.

So, what's the theory of knowledge of the theory of knowledge? — Terrapin Station

The question is: What is the knowledge-justification method in epistemology? Pattern matching, just like in science, but instead of matching them to real-world phenomena, it matches them to knowledge statements.

There may be no such thing as 'objective knowledge'. — fresco

That really depends on the definition of the term "objective". If the degree of objectivity of a shared belief increases with the number of believers, we can actually measure objectivity.

Objective does not mean and should not mean "true". It also does not mean and should not mean "justified". A completely unjustified belief can perfectly be very objective.

I think that it is not a good idea to commingle the terms "true", "justified", and "objective". In my opinion, the term "objective" does not necessarily mean "better".

The deconstruction of the subjective/objective dichotomy is prominent issue in epistemology.. — fresco

That is indeed why its definition is an essential issue.

Still, I am mostly interested in the justification of shared beliefs.

There is no case in physics, or in any other science, in which observations logically imply a theory. Observations are particulars, while theories make universal claims. — Dfpolis

The theory (Q) is justified by its experimental test reports (P). Therefore, P => Q.

The universal claim is not justified by visiting all cases in the universe. It is justified by visiting a mere sample. That is also why Q is not provable, as a counterexample cannot be excluded. That is the essence of the scientific method.

The portion of mathematics following from propositions abstracted from nature, or testable by observation (e.g. the parallel postulate), is scientific. The portion deriving from unfalsifiable hypotheses (e.g. the axiom of choice) is clearly not scientific, for it violates the accepted canons. — Dfpolis

Mathematics has its own canons. Science is one epistemic method and mathematics is another. Seriously, if your only tool is a hammer, then the entire world will soon start looking like a nail.

our capacity to investigate those foundations shows that being an axiom does not preclude justification — Dfpolis

That would only lead to infinite regress. Therefore, this approach is rejected in mathematics. As Aristotle said: "If nothing is assumed, then nothing can be concluded."

We can prove it by (1) noting that apples and pears are both fruit, (2) that they are also both units, and (3) applying ordinary arithmetic via the dictum de omni. Feel free to rebut this. — Dfpolis

Maybe you should read the basic instructions of Oregon State University for freshmen novice students:

What "proof" means in everyday speech:

In casual conversations, most people use the word "proof" when they mean that there is indisputable evidence that supports an idea.

Scientists should be wary of using the term "proof". Science does not "prove" things. Science can and does provide evidence in favor of, or against, a particular idea. In science, proofs are possible only in the highly abstract world of mathematics.

What should scientists say instead of "proof"?

Scientists should use the term "evidence" instead of the word "proof". When we test our hypotheses, we obtain evidence that supports or rejects the hypotheses. We do not "prove" our hypotheses.

While this may seem like a subtle difference, the words we use can subconsciously color our thinking. "Proof" suggests that a matter is completely settled, that we have had the last word on something.
...
In this class, therefore, I will ask you all to be mindful of using the term evidence rather than proof.

You seem to have missed the very basic training that was supposed to teach you not to use the term "proof" outside axiomatic derivation in mathematics.

It looks very much like the Oregon State University would disqualify you, and bar you from calling yourself a scientist.

In fact, that is a generalized problem with scientism. The worse the scientific training, the more the person becomes prone to the problem:

Scientism is an ideology that promotes science as the only objective means by which society should determine normative and epistemological values. The term scientism is generally used critically, pointing to the cosmetic application of science in unwarranted situations not amenable to application of the scientific method or similar scientific standards.

As I have argued already, mathematics obviously has its own normative and epistemological values.

This is irrational and inconsistent. You claim that mathematics need not be justified by observation. I hope you would agree that it is a mathematical truth that 2 + 2 = 4. — Dfpolis

The statement "2+2=4" is trivially provable from Dedekind-Peano's axiomatization of number theory. Still, the fact that the statement is provable in the abstract, Platonic world of number theory does not necessarily make it correspondence-theory "true" in the real, physical world.

In fact, that is not even possible, because the numbers "2" and "4" are an abstract language objects that do not appear in the real, physical world. You can also call them "two" and "four", or "deux" and "quatre". These things are not real-world objects. They are language expressions. Since when do language expressions have physical attributes? How can something be part of the real, physical world without any physical attribute at all?

They merely work out the implications of axioms that may or may not be justified by our experience of the real world. — Dfpolis

The implications, i.e. theorems, are exclusively justified from necessarily following from the axioms. It has nothing to do with the real world. The axioms themselves are never justified. Otherwise, they would not be axioms, because their justifications would then instead be the axioms. That way of thinking obviously just leads to infinite regress. Hence, justification of axioms is a fruitless activity.

If the axioms are justified — Dfpolis

Justifying axioms is exactly what does not make sense for mere procedural reasons. If you must justify the axioms, why would exempt you from also justifying their justifications? That approach leads to infinite regress, and is therefore not viable.

If the axioms are unfalsifiable hypotheses, those applying them are merely playing complex mental games. They are entitled to play their favorite games, but they can hardly expect society to support their play. — Dfpolis

The Stack Exchange question How does one justify funding for mathematics research? undoubtedly gives a reasonably good overview of why there is quite a bit of funding for reality-divorced, pure mathematical research.

One large and growing source of funding over the 20th century have been the military and intelligence departments. For example, you cannot do strong cryptography at any reasonable level without developing elaborate seemingly unrelated theorems in pure number theory.

In the link, you can see what kind of government departments and agencies subcontract research in mathematics. These grants are obviously not for plucking low-hanging fruit, such as slavishly mirroring reality.

Another funding source has been companies like IBM, who may rather be interested in fundamental computer science but often ends up dabbling in, and publishing pure mathematics. For example, Elsevier, a large academic publisher also has grants for research in mathematics.

Seriously, there is quite a bit of funding for reality-divorced research in pure mathematics.

And 'reasons for knowing' can operate at both an individual and a social level. — fresco

Yes, I was only dealing with standard, objective methods for knowledge justification. It may be possible that knowledge is subjective. A belief may be knowledge for a particular individual, because he can justify it, but other people may not. I was actually only looking into shared beliefs, i.e. objective ones.

Basically they are immigrants who come mainly from Libia, where there is a civil war. — Patulia

I remember French fighter jets bombing Khadaffi's troops in order to make sure that the rebellion would succeed. It is not that I particularly liked Khadaffi, but shouldn't France take in these immigrants instead of Italy?

On your account, mathematics is no more that a game -- not any different from Dungeons and Dragons, which also has rules that are neither true nor false, but simply to be followed by those playing the game. Funding mathematical research would be a scam in which we are paying people to play arbitrary games, with no hope of advancing our knowledge of reality, however theoretical. — Dfpolis

Hardy already admitted exactly that, in 1940, in "A Mathematician's Apology". It is not a secret:

Hardy preferred his work to be considered pure mathematics, perhaps because of his detestation of war and the military uses to which mathematics had been applied. He made several statements similar to that in his Apology:

I have never done anything "useful".

No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.

Hardy regards as "pure" the kinds of mathematics that are independent of the physical world.

Or even:

We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.

Furthermore, the low-hanging fruit is only moderately useful. It is rather some of the hard, abstract stuff, that initially looks useless, even for centuries, that will eventually turn out to be a real game changer. For example, Euler's centuries-old work on number theory, became a multi-trillion dollar business when Rivest-Shamir-Aldeman (RSA) kicked off the world of public-key cryptography.

Seriously, it is not the quick wins that make humanity progress.

My preferred approach is to use an analogous definition of truth as adequacy to the needs of a particular discourse. Then, for example, Newtonian physics is true with respect to many engineering needs. — Dfpolis

Concerning the coherence theory of truth, I agree with Bertrand Russell's objections:

Perhaps the best-known objection to a coherence theory of truth is Bertrand Russell's. He maintained that since both a belief and its negation will, individually, cohere with at least one set of beliefs, this means that contradictory beliefs can be shown to be true according to coherence theory, and therefore that the theory cannot work. However, what most coherence theorists are concerned with is not all possible beliefs, but the set of beliefs that people actually hold. The main problem for a coherence theory of truth, then, is how to specify just this particular set, given that the truth of which beliefs are actually held can only be determined by means of coherence.

Therefore, I cannot agree with "Newtonian physics is true with respect to".

Recall that the root meaning of "geometry" is "land measure" and many of its axioms are true of real-world geometric relations. — Dfpolis

You would have to visit all possible planets in the universe in order to verify that they are true of real-world geometric relations. You cannot do that. You will only sample some of these. Therefore, you cannot exclude the existence of counterexamples. Hence, these axioms are neither provable nor true about the universe.

We also know some of the principles of real-world existence. No real thing can be and not be in one and the same way at one and the same time, and so on. — Dfpolis

Entanglement allows for simultaneous being and not being in the real world. Schrödinger's cat is another example. Therefore, nuclear physicists seem to beg to disagree with you.

We can measure the interior angles of plane triangles and see if the results agree with the prediction that they will sum to two right angles. Then, the result is both axiomatically derived and experimentally confirmed. — Dfpolis

You cannot visit all possible such angles in the real, physical world. Therefore, the theorem is not provable about the real, physical world. It is only provable in the abstract, Platonic world in which the provability of this theorem is the result of the construction logic of that abstract, Platonic world. You can perfectly-well visit all such angles in an abstract, Platonic world. That doesn't cost energy. In the real, physical world, you would need more energy that you could ever practically amass.

I hate to break it to you, but there is no Platonic world. There is the real world and there are mental constructs that exist in the minds of people living in the real world. — Dfpolis

These mental constructs are abstract, Platonic worlds. They are not real. They are called Platonic because they are very similar to Plato's forms (but not necessarily the same):

Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. The term Platonism is used because such a view is seen to parallel Plato's Theory of Forms and a "World of Ideas" (Greek: eidos (εἶδος)) described in Plato's allegory of the cave: the everyday world can only imperfectly approximate an unchanging, ultimate reality.

A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is the Ultimate Ensemble, a theory that postulates that all structures that exist mathematically also exist physically in their own universe.

Platonism is the dominant philosophical view in mathematics.

Historically, most axioms have been abstracted from our experience of reality. — Dfpolis

Originally, yes.

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.[1][2][3] Two of the most highly abstract areas of modern mathematics are category theory and model theory.

However, this is no longer the dominant source of inspiration for axiomatization.

For example, the lambda calculus has absolutely no origins in the real, physical world. Nor do the various combinator calculi. None of Stephen Kleene's work, such as his famous closure, have any origin in the real world.

The entire discipline of computability has no connection, and has never had any connection, with the real, physical world. There is absolutely nothing that looks like a shift-reduce parser in the real world.

The mathematical foundations of computer science have never been about mimicking the real, physical world. That would simply be an exercise in futility. A running process on a computer system creates a virtual world of which the nature is studied using Platonic abstractions. Suggesting that these virtual worlds have originally been abstracted away from the real, physical world, is absurd. They do not exist in the real, physical world.

Alan Turing's Halting problem is provable in the abstract, Platonic world of running processes. What is the link with the real, physical world? In what way does the real, physical world contain running processes? Where are the naturally-occurring CPUs and computer systems?

Seriously, mathematics transcends the real, physical world. Physicists are just one group of its users. I do not understand why they think that they would be so privileged in connection with mathematics? Historically, there used to be an empirical link, but that link has been abstracted away a long time ago. There is no 20th century mathematics that still has such link. Mathematicians do not desire such link, because it would hold things back. Such link is very, very retrograde. Ever since the axiomatization of set theory in 1905 by Zermelo and Fränckel, absolutely nobody still wants that link.

Those of us trained in the natural sciences do not see unfalsifiable as an advantage, especially given that Godel work ruling out consistency proofs in systems representable in arithmetic. — Dfpolis

Well, ... in systems of which the associated language is capable of expressing the axioms of arithmetic.

Gödel was talking about the minimum power of a virtual machine and what we would today call its bytecode instructions. If the bytecode language can express Dedekind-Peano, the language can express (logical) truths that are not provable in the system.

Gödel's incompleteness is a language problem. The language required to express the axioms is more powerful than strictly what the axioms express. It is this fundamental mismatch that causes the problem.

In fact, Tarski's undefinability theorem is much better at expressing what Gödel's conundrum entails:

Smullyan (1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by Gödel's incompleteness theorems. That the latter theorems have much to say about all of mathematics and more controversially, about a range of philosophical issues (e.g., Lucas 1961) is less than evident. Tarski's theorem, on the other hand, is not directly about mathematics but about the inherent limitations of any formal language sufficiently expressive to be of real interest. Such languages are necessarily capable of enough self-reference for the diagonal lemma to apply to them. The broader philosophical import of Tarski's theorem is more strikingly evident.

As a passing note, the axiomatic method does not work for morality, nor did Aristotle claim that it did. — Dfpolis

It does work for morality. According to Kant's Critique of Practical Reason, the core of a moral system are its categorical imperatives, i.e. its axioms. In fact, Socrates already suggested that: "The understanding of mathematics is necessary for a sound grasp of ethics."

If they cannot be tested, they are unfalsifiable hypotheses and highly suspect. — Dfpolis

That is the empirical view in science, but a constructivist heresy in mathematics:

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. Even though most mathematicians do not accept the constructivist's thesis that only mathematics done based on constructive methods is sound, constructive methods are increasingly of interest on non-ideological grounds.

If your only tool is a hammer, then the entire world will end up looking like a nail. If you cannot transcend the sandbox of physics, you will misunderstand and mismanage everything you ever do, outside physics. Is it so unthinkable to you that other epistemic methods are different from your own? Doing mathematics in the way you suggest, is simply not mathematics. It would be a failed form of physics.

As we have just agreed, the original justification of mathematical axioms was not via deduction from more fundamental assumptions, but via abstraction from reality. — Dfpolis

Only pre-20th century mathematics mostly originated via abstraction from reality. However, most of the progress that has been booked after that, does not.

Why do we care? Because mathematics is a science -- as one organized body of knowledge among many. So, we want its conclusions to be true. In fact, truth is a central issue in Goedel's work. The problem he exposed (which completely undercuts your position) is that there are true theorems that cannot be proven from fixed axiom sets. If mathematics did not deal with truth, this could not be the case. — Dfpolis

Gödel does not talk about correspondence-theory "true". You can even trivially understand that from his canonical example:

S = "S is not provable in theory T"

Is S provable in T? No, because that would be a contradiction. Hence, S is (logically) true. Therefore, we are now sitting on a theorem that is (logically) true but not provable.

The language L associated with T is powerful enough to express S, and therefore, S is a relevant theorem in T.

The undefinability theorem does not prevent truth in one theory from being defined in a stronger theory. For example, the set of (codes for) formulas of first-order Peano arithmetic that are true in N is definable by a formula in second order arithmetic. Similarly, the set of true formulas of the standard model of second order arithmetic (or n-th order arithmetic for any n) can be defined by a formula in first-order ZFC.

In other words, it will be "true" and not provable in T, but it will be provable in any theory of which T is a sub-theory. The real, physical world is not chained into this tower of theories. Hence, it has nothing to do with correspondence-theory "true".

Further, if the truth of P is indeterminate, so is the truth of Q if its sole justification is P => Q. — Dfpolis

Yes, mathematical theorems are not correspondence-theory "true". They are only provable in their abstract, Platonic world.

So, the axiomatic method does not, and cannot, provide us with an exhaustive inventory of mathematical truths. That means that it cannot be the foundation of mathematical truth as you seem to imply. — Dfpolis

There are no mathematical truths. There are only theorems provable from the construction logic of their abstract, Platonic world, i.e. their axioms.

On your account, mathematics is no more that a game -- not any different from Dungeons and Dragons, which also has rules that are neither true nor false, but simply to be followed by those playing the game. — Dfpolis

Agreed.

Funding mathematical research would be a scam in which we are paying people to play arbitrary games, with no hope of advancing our knowledge of reality, however theoretical. — Dfpolis

There is no hope of advancing our knowledge of reality through mathematics. In relation to theories about the real, physical world, mathematics only supplies a consistency-maintaining bureaucracy of formalisms. Physics uses these formalisms. Hence, mathematics is useful to physics.

Finally, it mathematics were not true, it would not be applicable to reality. — Dfpolis

Mathematics is not applicable to reality. You will have to use another discipline for that purpose. You may indeed encounter mathematics as a tool to maintain consistency in what this other discipline claims, but that does not mean that mathematics would say anything about the real world.

Physicists who included mathematical premises in their reasoning, would be relying on claims of questionable or indeterminate truth, making their own conclusions and hypothetical predictions worthless — Dfpolis

Physicists do not include mathematical premises in their reasoning. They only maintain consistency in their theories by using mathematics. That works like a charm.

There are text books with numbers, that would never pass as reasoned texts if you took the numbers out of it. Same with diagrams in textbooks. — god must be atheist

Numbers are still language. Diagrams are not.

Bourbaki sought to ensure the purity of mathematics by axiomatizing and algebraizing. Bourbaki does not make explicit reference to Kant, but reaches the same conclusions as Kant about the use of visual illustrations:

It is fairly clear that the Bourbaki point of view, while encyclopedic, was never intended as neutral. Quite the opposite: it was more a question of trying to make a consistent whole out of some enthusiasms, for example for Hilbert's legacy, with emphasis on formalism and axiomatics.

Furthermore, Bourbaki makes only limited use of pictures in their presentation. Pierre Cartier is quoted as later saying: "The Bourbaki were Puritans, and Puritans are strongly opposed to pictorial representations of truths of their faith." In general, Bourbaki has been criticized for reducing geometry as a whole to abstract algebra and soft analysis.

While several of Bourbaki's books have become standard references in their fields, some have felt that the austere presentation makes them unsuitable as textbooks.

Of course, even though pictorial representations may be legitimate to aid understanding, it is necessary to prevent students to involve them in formal proof; which must be text only. Therefore, I do support Bourbaki's iconoclasm.

However, your post did not take anything away from my criticism of his finding that pure reason must be lingual only. — god must be atheist

Statements of reason, pure or not, will have to be expressed in language. Otherwise, they cannot be communicated.

The term "purity" in Critique of Pure Reason means: no sensory input. Accepting visual, auditive, tactile, smell, or any other such signal in the process of producing a theorem, is not allowed.

Kant already clarified that using a straightedge and compass, as advocated in Euclid's Elements, to draw geometric figures, and then solve visual puzzles in them to produce a theorem, simply amounts to accepting sensory input, and is therefore not to be considered pure reason.

Nowadays, we have an alternative approach to geometry that is fully algebraic, i.e. language only, and that therefore successfully addresses Kant's objections.

Is there anything else that you can use, besides language, that will properly stay clear of accepting real-world input? In my impression, there isn't.

There's no point in being bound to reality. — Dfpolis

The real, physical world is something that will always be systematically eliminated from mathematics:

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

In mathematics, the real, physical world is treated as an unwanted impurity that needs to be cleaned away, until it is finally gone. Good riddance!

The underlying error in most quantum mythology is Whitehead's fallacy of misplaced concreteness. It consists in treating abstractions as though they were actual, contextualized reality. The more formal and abstract, the more axiomatic, one's thinking, the more prone one is to this fallacy. — Dfpolis

Axiomatic thinking is meant to be used for abstract, Platonic worlds. It is not a tool for justifying statements about the real world, which are supposed to be backed by experimental testing.

There's no point in being bound to reality. — Dfpolis

In mathematics, indeed, no.

And the experience of being as quantifiable, from which to abstract the relevant concepts. — Dfpolis

Mathematics does not seek to be an abstraction of the real world. That is what physics seeks to be.

Clearly, we may not believe (accept) what we know, which would be impossible if knowledge were a species of belief. — Dfpolis

If you know it, it means that you can justify it. So, why would you not believe it?

If you do not assume that the axioms are true, then one cannot assume that anything derived from them is true. — Dfpolis

Mathematical theorems are not "true" in a correspondence theory (CT) sense. Theorems are merely provable from axioms, which themselves are never "true" in a correspondence theory (CT) sense.

Provable is never CT true --> because a provable theorem is part of an abstract, Platonic world
CT true is never provable --> because we do not have the axioms of the real world, i.e. the ToE.

Hence, CT true and provable exclude each other.

If we only need begin with unjustified axioms, we can start with any assumptions and prove anything. — Dfpolis

No. A system becomes trivialist because it contains a contradiction, for example.

In my view, axioms can be justified by abstraction, and most mathematical axioms are. — Dfpolis

Axioms do not need to resemble the real world in any way. That is simply not a requirement. Axioms can best be considered to be arbitrarily chosen.

If such measurements did not confirm this prediction, we would reject the parallel postulate -- as we do for non-Euclidean metrics. — Dfpolis

Justification by experimental testing, is not mathematics, but physics. Math does not justify axioms by experimental testing. In fact, Math does not justify axioms at all. If you justify axioms by experimental testing, then it is simply not math. In that case, you are doing something else.

Since we know the axioms are true any valid deduction from the axioms must be true (logic is salve veritate.) — Dfpolis

Axioms are not correspondence-theory true, and any theorem proven from such axioms is not correspondence-theory true either.

Mathematics does not seek to model the real, physical world. Physics tries to do that. Mathematics and physics are different disciplines that are in many ways diametrically opposed.

I personally do not believe that a good physicist could ever be a good mathematician, nor the other way around. That what is mandatory in the one, is strictly forbidden in the other. The one's virtues are the other one's heresies.

Changing hats is really hard, because if you wear the same hat, day in day out, it becomes a second nature. The more advanced you become in the one, the less suitable you become for the other.

Sets are well-defined collections of distinct objects, while accepted science is indeterminate because it varies over time. — Dfpolis

Science is a growing collection of theories that can be justified by the scientific method. At any point in time it is a set, but over time it is a changing collection. I concede that sets are immutable. In fact, I only wanted to refer to the fact that scientific theories are enumerable.

What defines a science is what is studied (its material object) and the approach to studying it (its formal object) — Dfpolis

That is probably true for "a science" but not for "science", which is simply any proposition that can be justified by experimental testing.

Also, while I agree that scientific findings can be justified beliefs, in many cases we have no way to determine whether they are true in any absolute sense. So, your definition of knowledge cannot be applied to scientific findings in general. — Dfpolis

Yes, agreed. I do not think that knowledge is necessarily a "true" belief, with the term "true" as in the correspondence theory of truth. Knowledge as a "justified belief" should be sufficient.

Why is that? — Dfpolis

Math justifies by axiomatic derivation, while science is does that by experimental testing. Experimental testing always occurs in the real, physical world, of which we do not have the axioms. Therefore, we cannot axiomatically derive that what can be experimentally tested. The converse is also true. If a proposition is derived axiomatically from a set of axioms that construct an abstract, Platonic world, you cannot experimentally test it, because that would require the objects to be part of the real world and not the Platonic world in which they have been constructed.

I do not see that there is an axiomatic method. — Dfpolis

The axiomatic method is defined and discussed in numerous places, such as here and here.

In mathematics, the axiomatic method originated in the works of the ancient Greeks on geometry. The most brilliant example of the application of the axiomatic method — which remained unique up to the 19th century — was the geometric system known as Euclid's Elements (ca. 300 B.C.).

After Euclid's Elements introduced the axiomatic method, Socrates got the idea that philosophy had to be approached in a similar manner. The approach did not entirely succeed, and it was not a good idea for science, as would later become clear from Aristotle's now outdated scientific publications, but it works for mathematics and morality.

Axioms can be abstracted from reality — Dfpolis

That is how axioms were originally understood:

Axiomatic method, in logic, a procedure by which an entire system (e.g., a science) is generated in accordance with specified rules by logical deduction from certain basic propositions (axioms or postulates), which in turn are constructed from a few terms taken as primitive. These terms and axioms may either be arbitrarily defined and constructed or else be conceived according to a model in which some intuitive warrant for their truth is felt to exist.

Nowadays, axioms are deemed to be arbitrarily defined. They are unrelated to the real, physical world. Axioms are not true nor false, in terms of the correspondence theory of truth. Axioms are the building bricks of a new, abstract, Platonic world.

How does the so-called "axiomatic method" justify its axioms? — Dfpolis

It doesn't. In fact, that is even forbidden, because in that case, they are not axioms. In a knowledge statement P => Q, you can see that Q is justified by P. We do not care how P is justified, or if this is even the case. It does not change anything to the fact that Q necessarily follows from P. The knowledge expressed by P => Q, as a justified belief, is not P nor Q, but the arrow between both.

Similarly, the scientific method takes takes observations and the logic involved in working out the implications of hypotheses as given, and provides no justification for either. — Dfpolis

In science, the observations are the P (justifying statement) and the theory (knowledge statement) is the Q, in P => Q:

(P) justifying statement => (Q) knowledge statement

There is no need to justify P, because the status of P does not affect the arrow, which is the real knowledge.

If it were, mathematics would not be scientific. — Dfpolis

Mathematics is not justified by experimental testing, and is therefore, not scientific.

Second, it is an argumentum ab auctoritate from an unreliable source. Hawking, despite many admirable traits, has been wrong on fundamental matters much more central to his area of expertise. — Dfpolis

In his lecture, Gödel and the End of Physics, Hawking spent quite a bit of effort justifying his views. For me, it works.

While physics can be and has been axiomatized (e.g. quantum theory and quantum field theory) — Dfpolis

If it is physics, it is about the real, physical world, and in that case, you can test it. Therefore, it will not be accepted, as a matter of principle, that it does not get tested.

The justification for the axioms is that, so far, they seem to work. — Dfpolis

If "they seem to work" is about what they observed in the real, physical world. That amounts to again to testing.

So, a bowl that holds only one apple and one pear cannot be proven to hold two pieces of fruit? — Dfpolis

No. It will undoubtedly be true, but it will not be provable. To cut a long story short, there are numerous articles in the search results that explain why not.

All truths derive from experience. — Dfpolis

Yes, agreed, in the correspondence theory of truth, the real, physical world is the benchmark for truth.

So, 2 objects and 2 more objects might not yield a total count of 4 objects outside the visible universe? — Dfpolis

Doesn't matter, because you cannot observe it. Therefore, without observations in an experimental testing fashion, such claim about the non-visible universe is unscientific.

It does not matter that one can represent meta-mathematical relations mathematically, for if it did, mathematical physics would be subalterned to mathematics, and it is not — Dfpolis

Metamathematics is the axiomatic system about (other) axiomatic systems. It is an abstract, Platonic system that produces theorems about other abstract, Platonic systems. Mathematical physics, on the other hand, is still about the real, physical world. Therefore, it is not part of mathematics.

Mathematics requires you to painstakingly construct the world in which you will derive your mathematical theorems. We did not construct the real, physical world. Therefore, we are not allowed to derive mathematical theorems in it.

I agree and don't know why you went there. — Gnostic Christian Bishop

Your views on Islamic law reflect your ignorance on the matter. Furthermore, by shilling for the ruling elite, you completely lose credibility.

Kant was a ninnie. I've been harping that forever, but nobody pays me any attention. Instead, they turn to me and say, "how can you say that? BLASPHERMER!" Whereas all you have to do is read what Kant wrote, think about it for five minutes and you realize that the bloke was full of false views. — god must be atheist

I think that Kant is the greatest epistemologist ever to have set foot on this earth. I also consider him to be the first epistemologist to have made real progress after Plato and Aristotle. As far as I am concerned, after him, there are only Karl Popper and Edmund Gettier to have contributed meaningfully. Epistemology is a field with very few names to mention. There have been lots of philosophers but only a handful of them have managed to do something meaningful in epistemology.

With the above in mind, and given my neglect of the history of African-American disenfranchisement, why is it that we needed people like Martin Luther King, Malcolm X, and many others, which I am blissfully unaware of; but, shouldn't be, had to demand change? — Wallows

Dum Diversas (English: Until different) is a papal bull issued on 18 June 1452 by Pope Nicholas V. It authorized Afonso V of Portugal to conquer Saracens and pagans and consign them to "perpetual servitude".

The Atlantic coast of Africa being pagan, the Holy See authorized the consignment of these populations to perpetual servitude.

This is obviously a grave jurisprudential error.

In other religions, it works differently. After conversion, the parents may remains slaves, but the children will be born free.

This mishandling of slavery had ugly consequences. For example, it forced American president Thomas Jefferson to make great struggles to avoid consigning his children by Sally Hemmings to slavery. That is a depravity. You can obviously not enslave your own children, but that is an inevitable consequence of the Papal ruling.

There are gigantic jurisprudential issues in Christian morality which eventually culminated in the notorious exchange between Martin Luther and the Papacy at the Landstag in Worms, Germany, in April 1521.

Luther: If you can show me through scripture and reason that I would be wrong, I will retract what I have said.

Papacy: The Bible itself is the arsenal whence each evil heretic has drawn his deceptive arguments.

Later on, it turned out almost impossible to enfranchise people of African descent, as they had been consigned to perpetual servitude by the Papal bull, Dum Diversas.

Still, since the meek tend to inherit the earth, as exemplified during the Haitian revolution, the abolitionists urgently sought a way to avoid the otherwise inevitable from happening in the southern United States. Slave imports had to be stopped and slaves had to be freed, but it took a spectacular civil war to knock that understanding into the dumb skulls of the southern ruling elite.

Freeing the slaves was not a matter of sanctimonious virtue signalling, even though it was advocated in those terms, but one of long-term self-interest.

Christianity totally mishandled the issue of slavery, and has actually not been viable as a moral system for centuries now. The completely flawed Christian take on slavery is indeed an ugly exponent of this general problem.

Judaism has a Law (Hallakha) as its moral system, Islam too (Sharia), but Christianity has instead, its corrupt Church, i.e. the notorious Babylonian whore on whose forehead is written "mystery".

Are you advocating barbaric religious laws be the law of the land the way Muslims do? — Gnostic Christian Bishop

Islamic law has stiff maximum penalties for serious misbehaviour. Criminal justice is governed by sentencing guidelines. A maximum penalty of electrocution on an electric chair is not more lenient than one of decapitation. Furthermore, the maximum penalty is rarely if ever pronounced, and even then, tends to get commutated in times of peace. Last but not least, after having exhausted all options in the appellate procedure, it is still possible to request a pardon from the head of the state.

Seriously, arguing over maximum penalties is quite pointless.

Law is an endless bureaucracy of formalisms and procedures. Islamic law is not different in that respect. As ever, in those circumstances, you'd better have a good lawyer.

BTW, Americans fear their government while in other more advanced democracies, the governments fear the people. American have lost the decent balls they used to sport. — Gnostic Christian Bishop

The population must at all times reserve the option to overthrow the existing regime. When the state faces an individual or a small group, they can use legal means. That does not work if a group of even just 1% of the population decides to take them on. Then, it is group against group, and hence, politics. Political conflict cannot be decided by the law, because the very question of politics is exactly what the law should be. Hence, in political conflict we are absolutely not bound by the ruling elite's secular law-concoctions. Their invented laws do not matter, because the very purpose of the political conflict is to get rid of them as well as their concocted laws.

We do not idol worship anything or anyone. — Gnostic Christian Bishop

And God said: "I am the lawmaking God who freed you out of slavery in Egypt. One. First law. You will recognize no other law makers but me."

Through your glorification of the ruling elite's secular-law concoctions, you have associated the ruling elite as law making partners to God, or even above God. That puts you in violation of the first law transmitted by Moses. Therefore, you effectively are an idol worshipper.

You are even a dangerous idol worshipper. Your false gods also want to impose their secular-law concoctions onto those believers who do not wish to break God's first law. Hence, you also lend material support to the enemies of the believers. Seriously, there is no difference between you and the enemy.

There was no Allah to speak of, for example, until the advent of the Islamic religion, regardless of whether He existed prior to that ... — Maureen

Islam emerged out of Messianistic Judaism, i.e. Ebionite Christianity.

That is why Hans Joachim Schoeps wrote:

Thus we have a paradox of world-historical proportions, viz., the fact that Jewish Christianity indeed disappeared within the Christian church, but was preserved in Islam and thereby extended some of its basic ideas even to our own day. According to Islamic doctrine, the Ebionite combination of Moses and Jesus found its fulfillment in Muhammad.

In fact, Ebionite Christianity split off from Second-Temple Judaism before even Rabbinic Judaism did. The name "Allah" refers to the God of Second-Temple Judaism, i.e. the God of Moses.

Second-Temple Judaism itself disappeared after the destruction of the Second Temple in 70 A.D during the first Jewish-Roman War.

The mind uses, then, induction upon basic universal axioms (mathematical and logical) to conclude the existence of various highly complex (contextual) metaphysical cause and effect rules, which lead to the conclusion of the existence of an objective substance that is the source of perception and knowledge. — Nasir Shuja

We do not have these basic universal axioms. We have no access to the Theory of Everything (ToE), also called, the Preserved Tablet of Wisdom. If we did, we would be able to flawlessly predict the future.

Therefore, the mind must necessarily use something else, which is less complete and less consistent. Humanity may use more than mere knowledge -- I also believe that -- but it is clearly not anything like perfect knowledge.

Religions tout themselves as being the final word in moral issues, even though secular law has rejected as too barbaric most of those laws. — Gnostic Christian Bishop

Secular law is a concoction by the ruling elite at whose core core you can find a bunch of corrupt banksters. Of course, it suits the ruling elite absolutely fine that you seek to glorify their inventions while seeking to discredit alternatives.

Unlike you, I do not trust the ruling elite.

Therefore, I have a vested interest in countering the ruling elite with any alternative available, including religion. If sometimes religion can be the tool of the ruling elite, it can also trivially be repurposed into a tool against it.

To me, a moral god would cure and never kill. — Gnostic Christian Bishop

Well, you happily accept that a policeman hits you with a gummy stick, but you would object to religion doing that. That is because you are a worshipper of the ruling elite. They are your gods, but they are certainly not mine.

The power of the ruling elite existentially depends on the idea that they can hit you, but you cannot hit them back. It is the core of your morality. At the core of my morality, you will find the opposite view: Hitting them back is useful, interesting, and very necessary. You glorify violence by the ruling elite while I glorify violence against the ruling elite.

When a war breaks out, the very first job always consists in singling out any possible defeatists, round them up, and to promptly terminate them. When a revolution breaks out, the very first job consists in singling out the shills of the ruling elite, round them up, and then extensively decimate them.

Seriously, attacking religion with a view to prop up the ruling elite and their secular-law concoctions is absolutely not free of charge. You can do what you want, but so can we.

This idea doesn't work for one fundamental reason: once you have depleted a mathematical proposition of any meaning, you have no clue why that theorem should be true, or should be distinguished from the infinite sea of combinations of symbols that can be interpreted as theorems. — Mephist

There is also a merely mechanical reason why it does not work: Gödel incompleteness theorems. I am actually not against the use of meaning, i.e. informal semantics, in mathematics. I am only against the use of semantics as proof; which should be syntactic only.

Actual meaning will be plugged in by the discipline that applies the mathematics.

For example, in physics, the semantics concern a deeper understanding of the real, physical world. Therefore, physics is supposed to be semantically oriented and semantics-heavy. If people are interested in understanding the real world, they'd better do something like physics, and not pure mathematics.

you cannot teach mathematics as a purely symbolic game, because in this way it has no meaning at all. — Mephist

Agreed. There is an important difference between teaching versus proving mathematics. Even though semantics are banned from being used in proofs, they are actually ok elsewhere. I only objected to V.I. Arnold suggesting that the epistemic justification method in math should be empirical.

In other words: the meaning of a theory is not contained it it's purely symbolic representation, but in it's correspondence with the way the physical world works. In this sense, algebraic geometry (for example) is not substantially different from Maxwell's equations. — Mephist

There may be a danger in encouraging people to look for semantics/meaning in mathematics itself instead of picking a physics theory such as Maxwell's equations.

(Pure) mathematics is for people who appreciate the beauty of formalisms and surprising structures that emerge out of the design of abstract, Platonic worlds. It really is a goal in itself, and it is supposed to be done with total disregard for possible applications.

It is a particular type of sensitivity and talent that some people have and others not really.

Why would everybody need to study subjects that they may not even like?

There are so many other fields of endeavour where people can find satisfaction in pursuing understanding. I think that it is wrong to force everybody to learn more than just basic arithmetic, just like it would be wrong to force everybody to every day spend hours in learning how to play the violin.

It is true that from any set of axioms you can build a theory and derive the relative theorems, but I guess that nobody would be interested at all in axioms that do not correspond to any generalization whatsoever model that corresponds to ideas taken from the physical world. If you choose the 'wrong' axioms, you obtain a meaningless theory. — Mephist

Well, we actually do it all the time.

For example, Chris Barker's Iota combinator calculus is a Turing-complete system with just one combinator, i.e. Iota. People already wondered if the SK combinator calculus could be simplified from two symbols to one. So, Chris Barker positively answered that question.

I always thought that two symbols was the minimum, but now we know that just one is enough.

By the way, combinator calculus is undoubtedly "useless" in a sense of not having any applications. To tell you the truth, I actually prefer systems that do not readily have applications waiting for them. If these applications are eventually found anyway, they tend to be extremely powerful. Low-hanging fruit, on the other hand, tends to be rather worthless on the long run.

By the way, a "combinator" has no meaning. It is just a symbol in the system. I like it that way, because that kind of symbols should have no meaning. Combinator calculi are indeed a collection of meaningless theories. I like it that way.

Kleene's closure is an example of a theory that was utterly useless for a very long time, but surprisingly beautiful, and even intriguing. In the meanwhile, it has turned into a rage, some kind of hype. Almost every non-trivial piece of software now uses it. The following is a popular portal for regular expressions, and this is an online testing tool for RE.

Regular expressions (RE) have no "meaning". They are just syntax patterns. Still, they often trivially solve otherwise really big problems in software. If RE actually had a meaning, they would not even be useful ...

This is a citation from "On teaching mathematics", by V.I. Arnold — Mephist

V.I. Arnold is a constructivist heretic.

I had to cringe when reading his article.

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry.

Mathematics supplies its consistency-maintaining bureaucracy of formalisms to more than one other field. It is not just the private prostitute or concubine of physics. What about people in engineering, business, computer science, and industry?

Mathematics is a part of physics.

You see !? The article already starts with designating mathematics as the private whore of physics.

Mentally challenged zealots of “abstract mathematics” removed all the geometry

Firstly, Immanuel Kant pointed out in his Critique of Pure Reason that the practice of solving visual puzzles, as in Euclid's Elements, could not possible be considered pure reason, because it rests on fiddling with visual input, while pure reason must be language only, entirely devoid of sensory input. That is one reason why an algebra-only, pure-reason approach to mathematics is much preferable to geometric fiddling with visual puzzles.

Secondly, Carl Friedrich Gauss algebraically described the limitations of geometrically constructing numbers with unmarked straightedge and compass:

Since the field of constructible points is closed under square roots, it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients.

In order to compute numbers that lie outside these quadratic field-extension towers we must use algebra. Geometry cannot handle such numbers, such as cube roots, or even roots of arbitrary power, and so on.

Geometrically constructible numbers are just a very, very small subset of all computable numbers. Therefore, these Euclidean methods hold us back. We had to drop them, in order to be able to progress.

The scheme of construction of a mathematical theory is exactly the same as that in any other natural science.

His constructivist heresies are nauseating!

Then we try and find the limits of application of our observations by seeking counter-examples ...

That is physics. That is not math!

certain facts which are only known with a certain degree of probability or with a certain degree of accuracy, are considered to be “absolutely” correct and are accepted as “axioms”

Axioms are not "correct". Axioms are just arbitrary starting points for the construction of an abstract, Platonic world.. Axioms have nothing to do with the real world (just like everything else in math).

The mathematical technique of modelling consists in ignoring this trouble and speaking about your deductive model as if it coincided with reality.

Mathematical systems (not "models") are not even meant to coincide with reality. Mathematics never says anything at all about the real world.

His views are heretical!

Attempts to create “pure” deductive-axiomatic mathematics have led to the rejection of the scheme used in physics (observation, model, investigation of the model, conclusions, testing by observations) and its replacement by the scheme definition, theorem, proof. It is impossible to understand an unmotivated definition but this does not stop the criminal algebraist-axiomatizers.

And he is a criminal constructivist!

Axioms are unmotivated definitions because any requirement to motivate them would lead to infinite regress. As Aristotle said: "If nothing is assumed, then nothing can be concluded."

Any attempt to do without this interference by physics and reality with mathematics is sectarian and isolationist, and destroys the image of mathematics as a useful human activity in the eyes of all sensible people.

By shedding Euclid's Elements, mathematics has finally become pure reason, which is language (symbol manipulation) only. We do not want so-called usefulness. We want purity, because ultimately, it is purity that is math's usefulness.

... university mathematics courses (from which in France, by the way, all geometry has been banished in recent decades).

Yes, the impurities of visual puzzling had to be stopped. Good riddance. Math had to become pure reason, i.e. language only.

If mathematicians do not come to their senses, then the consumers, who continue to need mathematical theory that is modern in the best sense of the word and who preserve the immunity of any sensible person to useless axiomatic chatter, will in the end turn down the services of the undereducated scholastics in both the schools and the universities.

Math is staunchly axiomatic. That is non-negotiable. Seriously, get over it!

Someone had better tell this author to come to grips with the fundamental epistemic method of math; and if he does not like it, then he should stick to physics instead.

Thus, applied mathematics is a combination of mathematical science and specialized knowledge.

The term "mathematical science" is an oxymoron.

A theorem is either mathematics or science but can never be both simultaneously. The two epistemic methods are so diametrically opposed that it would not be possible.

In mathematics, claiming anything at all about the real world is a constructivist heresy.

Thanks, that's very instructive, but I think it's an artificial distinction. It glosses over most of what I find philosophically interesting about it. — Wayfarer

That is unfortunate, because the problem is caused at the level of epistemology, i.e. the theory of knowledge, which is in my opinion the most interesting subject in philosophy.

The existence of different epistemic methods gives rise to the existence of different epistemic domains: mathematics, science, and history. Other subjects may not even be powered nor delimited by an epistemic method, and therefore, have no standard justification method. Such subjects are therefore not even knowledge.

For example, what is the epistemic method of sociology or economics? If there isn't one, these subjects can be suspected to be mere conjectures.

The theory of knowledge is a powerful tool.

It suggests that 90% of what the academic world is doing, is not knowledge but just a haphazard collection of worthless conjectures. The national education systems are globally wasting trillions of dollars on teaching matters that are inherently worthless. Only the theory of knowledge is able to give us that important insight.

What about applied maths — Wayfarer

"Applied math" is not math.

If the object of mathematical language is the real world, it is not math. It is something else that merely uses mathematical formalisms to maintain consistency in its own statements, such as for example science.

what Eugene Wigner calls the 'unreasonable effectiveness of mathematics in the natural sciences' — Wayfarer

Yes, mathematics is unreasonably effective in maintaining the consistency of natural-science theories.

However, math is not natural science itself. As soon as you say something about the real world, it is not math, but something else.

On a really basic level, how come maths works in the real world, if it has nothing to do with it? — Wayfarer

Math is consistent by design. The real world is consistent by assumption.

If you use math in modeling the real world, mathematical consistency and assumed real-world consistency will tend to be isomorphic with each other. The semantics of the rules with which you model the real world will have to come from elsewhere than math, but the syntax will indeed be guarded effectively by math.

In that context, math is just a set of bureaucratic formalisms that prevents the real world modeler from contradicting himself or otherwise making inconsistent claims. However, the real world modeler will have to provide the semantic "meat" from a non-math source. Math simply does not provide semantic "meat" about the real world.

And we know that with a kind of mathematical certainty, because the mind knows mathematical truths, and the forms of things, with a far higher degree of certainty than it does mere sense impression. — Wayfarer

The term "mathematical truth", with the term "true" being defined by the correspondence theory (CT) of truth, is actually an oxymoron. A mathematical theorem will be "provable" from its axiomatic context, but never CT-true (about the real, physical world). Therefore, "provable" necessarily implies: not CT-true.

Mathematics is not CT-true by design.

Logically true (L-true) is also not CT-true, because L-true is just an arbitrary value in an algebraic lattice that represents a particular axiomatization of logic. Gödel's incompleteness proves the existence of knowledge that is L-true but not provable from any sufficiently-complex arbitrary axiomatic context. Hawking therefore says that this implies that there exists knowledge in the ToE that is CT-true but not provable from the ToE.

Traditionally, philosophers believed that it was the job of epistemology to justify our knowledge. In contrast, the central job of Quine’s naturalist is to describe how we construct our best theory, to trace the path from stimulus to science, rather than to justify knowledge of either ordinary objects or scientific theory.

Quine's naturalist's approach is actually epistemologically sound. If the purpose of some type of knowledge is to say anything about the real world, then it can only be justified by correspondence with the real world. A mathematical theorem does not seek to say anything about the real world, but only about its abstract-Platonic context, and is therefore entirely exempt from this requirement.

Hence, mathematics is not CT-true but is certainly provable.

The terms "true" and the term "provable" uncannily exclude each other.

Not all knowledge is about the real world (a posteriori). Kant already pointed that out by deriving the existence of synthetic statements a priori.

But naturalism is not at all critically self-aware in the sense that traditional philosophy actually was*. And it's also forgotten what it has excluded. This was one of the consequences of the attempt in the Enlightenment to discard metaphysics in favour of what was purportedly "really there", the so-called 'real world' as object of scientific enquiry. But in so doing, the West abandoned some essential and fundamental aspect of their intellectual heritage. — Wayfarer

Science, which is the exponent of naturalism, is by design, not possibly critically self-aware. The reason for this, is merely formal, and even purely mechanical.

We can define the term "science" as statements about (physical) observations, for which we can look for counterexample (physical) observations.

So, statements about (physical) observations (=science) can never be statements about statements about observations (=statements about science).

Therefore, statements about science are necessarily not scientific.

Hence, science cannot possibly say anything about itself, while such ability to say things about oneself is a prerequisite for critical self-awareness.

Could I suggest that in saying that, you're positing 'mind' as 'something within the individual' - my mind, or your mind, the conscious cognition of an individual human. Of course, within that picture, the individual is indeed only a phantasm. — Wayfarer

Yes, agreed. This problem is ignored and considered unimportant until our perception -- being an abstract model itself -- suffers from a serious abstraction leak, which inevitably, occasionally happens.

In the movie, The Matrix, taking the red pill even causes a permanent abstraction leak.

The individual in the blue-pilled world is, in fact, just a fantasy.

But then again, the world that they consider "real" after getting red-pilled actually has the same problems as the blue-pilled world that they then consider to be fake, the only difference being that they are not aware of that.

They never ask themselves the question if the red-pilled world is also not just a fantasy?

Not asking this question is a weakness in the movie. But then again, the audience which already has to get used to the idea of one fake world, could get badly confused by the idea of having landed in yet another fake world.

The audience watches all of that in a movie theatre screen which is specifically constructed to display pretty much fake worlds only, aka, "fiction".

the fact that it may use mathematical methods does not make it mathematics anymore than the fact that physics uses mathematical methods makes physics a branch of mathematics. — Dfpolis

Physics does not use the axiomatic "method". Physics uses mathematical formalisms to maintain consistency in its theories, but has actually nothing to do with mathematics. Maybe I did not express myself clearly. With the term "method", I meant "epistemic method", i.e. knowledge-justification method, as in axiomatic "method", scientific "method", and historical "method". I did not mean algebra or mere symbol manipulation. It was an epistemic concern only.

No, mathematics has quantitative relations as its subject matter — Dfpolis

Well, we will have to agree to disagree here.

Mathematics is not (Dedekind-Peano) number theory, which is no longer the dominant axiomatization nowadays. Contemporary mathematics defaults to Zermelo-Fränkel-Choice (ZFC) set theory as its dominant axiomatic context. The switch dates back to 1905. So, this has been the case for over a century now.

What we now call "metaphysics" was called "first philosophy" by Aristotle because it deals with issues fundamental to all other areas of research, including physics — Dfpolis

Metaphysics does not establish the epistemic method for any area or research, including physics. It is epistemology that does that job.

Mathematics is what you can justify using the axiomatic method, science using the scientific method, and history using the historical method.

That is not a metaphysical but epistemic concern.

Still, all three are sciences in the more traditional sense of rigorous systematic fields of study. — Dfpolis

According to Karl Popper's 1963 "Science as Falsification", which has in the meanwhile become the dominant view in the philosophy of science, science consists of the theories that you can justify by experimental testing. Mathematics never does that. Hence, mathematics is not science. Mathematics is not empirical and uses the axiomatic method instead. Therefore, it does not make sense to count mathematics under the nomer science. Furthermore, mathematics and science exclude each other. It is not possible to justify a theorem with both methods. It is the one or the other. Metamathematics is just a subdiscipline in mathematics, i.e. one particular axiomatization amongst many.

It is absurd to think that any competent physicist would accept a proposed ToE absent rigorous experimental testing. — Dfpolis

According to the late Stephen Hawking, the problem will never even occur. According to him, there simply won't be anything to test.

I am not sure an axiomatic approach is the "desired alternative" in natural science. You'd need to make a case for that. It seems to me that many physics like the method of discovery they signed up for. — Dfpolis

Well, the ToE is an axiomatic system, and physicists seem to dream of finding it. The late Stephen Hawking clearly did, but then he eventually concluded that it cannot be done.

I don't think that Hawking's view is widely shared. — Dfpolis

Well, I do not think that Hawking was infallible. He was just influential in his circles, and with a rare connection, say, even fascination from the general (layman) public. Especially because of his debilitating disease, people admired his tenacity, willpower and willingness to do some real work in spite.

metaphysics is concerned with being as being, which is not subject to the vagaries of measurement. — Dfpolis

Well, metaphysics seems to have very little influence nowadays on the practice of physics. This is not true for metamathematics, which thoroughly dominates the discourse in mathematics.

While I'm happy to admit that knowledge is a subject-object relation, I do not see that the admission precludes proofs about reality. — Dfpolis

It is otherwise an epistemic view widely shared by lots of mathematicians and scientists. If it is provable, then it is not about the real world. If it is about the real world, then it will not be provable. It harks back to the definition of the term "proof" as the derivation path between a theorem and its underlying axioms. Without axioms, no "proof".

A mathematical proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference.

You can clearly see that this is not possible in science. Science is backed by experimental testing. So, science cannot provide us with such derivation path to something that does not even exist in science, i.e. axioms.

I am sorry, but I do not see nature as an axiomatic construct. — Dfpolis

Well, you will always be right, because according to Stephen Hawking, nobody will ever be able to supply you with the axiomatic construct that will prove you wrong. The ToE is just an unattainable pipe dream anyway.

Rather, I see it as a complex, intelligible whole from which we may abstract some universal truths. — Dfpolis

Well, these "truths" -- I would rather say experimentally-tested "theories" -- have only been tested at best against observations in the visible part of the universe. I do not see how it would be possible to justify with observations anything about the non-visible part of the universe. Hence, it is very local knowledge, the visible part of the universe being deemed just a small fraction of the complete universe, most of which is merely being conjectured about.

At the simplest level, we understand being well enough to see that (1) Whatever is, is, (2) that a putative reality must either be or not be, and (3) that nothing can be and not be at one and the same time in one and the same way. — Dfpolis

Well, there seem to be physics theories that do not abide by this, such as Schrödinger's cat and the entire concept of entanglement.

But then again, these theories are too physical-world to my taste. That is why I do not know particularly much about them. I personally prefer the abstract, Platonic worlds of mathematics, for which you only need pen and paper. In fact, I actually, actively avoid physics, because I do not want to end up needing a gigantic particle accelerator or anything of the sort.

I reject the thesis that knowledge is any form of belief. — Dfpolis

The mainstream view is that knowledge is a justified (true) belief:

Justified true belief is a definition of knowledge that gained approval during the Enlightenment, 'justified' standing in contrast to 'revealed'. There have been attempts to trace it back to Plato and his dialogues.

This analysis precludes any knowledge, for it leads to an infinite regress (How do we know P?). — Dfpolis

P does not need to be knowledge. For example, axioms are not knowledge, because they are not justified. Therefore, there is no infinite regress in axiomatic knowledge. There is no infinite regress in scientific knowledge, because the final justification is provided by experimental test reports, which are historical knowledge, and justified by the historical method (Did they really take place?).

Also, I do not consider philosophy to be a closed axiomatic system — Dfpolis

I agree. It is obviously not. Otherwise, it would be mathematics. Still, you have to start somewhere. It will initially, and possibly even never, be possible to turn a philosophical idea into a rigorous system.

He does no such thing. He only considers closed, formal systems, not empirically open systems. If your claim were true, we would never have made progress in any science. — Dfpolis

Yes, of course. However, with access to the ToE -- which will never happen -- the distinction between axiomatic and empirical would disappear. Furthermore, knowledge of science is not enough to discover new science. There is another ingredient than mere knowledge that is needed for such discovery.

It is easy to show that the mind cannot be purely neurophysical. — Dfpolis

Agreed. If it were, it would not even work.

Interesting. What discipline is this from? Computer science? — Wayfarer

The lambda calculus was first described by Alonzo Church:

Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, proving the undecidability of the Entscheidungsproblem, Frege–Church ontology, and the Church–Rosser theorem. He also worked on philosophy of language (see e.g. Church 1970).

Alonzo Church did not have access to computers when he described the lambda calculus:

The lambda calculus was introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics.

There were no computers in the 1930s. Alonzo Church was known mostly as a mathematician and for his work in mathematics. There is a connection, however with what would later be termed "computer science":

The lambda calculus influenced the design of the LISP programming language and functional programming languages in general. The Church encoding is named in his honor.

Type theory was first described by Bertrand Russell.

Bertrand Arthur William Russell, 3rd Earl Russell, OM FRS[64] (/ˈrʌsəl/; 18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, essayist, social critic, political activist, and Nobel laureate.

His work has had a considerable influence on mathematics, logic, set theory, linguistics, artificial intelligence, cognitive science, computer science (see type theory and type system) and philosophy, especially the philosophy of language, epistemology and metaphysics.

Russell certainly did not have access to computers when he described his type theory:

Between 1902 and 1908 Bertrand Russell proposed various "theories of type" in response to his discovery that Gottlob Frege's version of naive set theory was afflicted with Russell's paradox. By 1908 Russell arrived at a "ramified" theory of types together with an "axiom of reducibility" both of which featured prominently in Whitehead and Russell's Principia Mathematica published between 1910 and 1913.

So, I disagree with the mention in his Wikipedia page that Bertrand Russell would have contributed type theory to computer science between 1902 and 1908. At that point in time, computers did not exist, not even on paper as a concept.

I just looked up philosophy in the dictionary _ a search for truth through logical reasoning rather than factual reasoning. — Corra

"Philosophy" is exactly the type of term that layman dictionaries tend to get completely wrong.

The person who wrote the definition, simply did not understand what the term means, and then proceeds to explaining it in terms of other terms that he does not understand either.

Wikipedia is usually better, but in case of the term "philosophy", it is just as bad. The reason why so much bad definition work is being produced, is that these people simply do not understand what the term "definition" means.

Therefore, let's have a look at the work of people who clearly do have a usable understanding of what the term "definition" means.

Imagine we want to teach how to visually detect if something is a banana. How do we train the device to do that?

For training a boosted cascade of weak classifiers we need a set of positive samples (containing actual objects you want to detect) and a set of negative images (containing everything you do not want to detect).

After the opencv_traincascade application has finished its work, the trained cascade will be saved in cascade.xml file in the -data folder. Other files in this folder are created for the case of interrupted training, so you may delete them after completion of training.

Training is finished and you can test your cascade classifier!

A cascade classifier file is obviously a definition. It allows the cascade classifier program to inspect images and decide whether it contains a banana or not.

So, what exactly is a philosophy definition?

It is a file that contains rules and/or a description that should allow the user of the file to classify statements into philosophical ones and statements that are not. If the definition does not allow for using it in that way, then it is not a legitimate definition.

Deciding whether or not to go to college is a major decision that people have to make in life. But, is it prudent for most people to go to college? — TheHedoMinimalist

No.

The first thing will be to pick a subject for your degree, while the very concept of subject as some kind of subdivision of the world of knowledge is highly nonsensical.

Let's pick an example: psychology, economics,and sociology. No matter what definition you pick for these subjects, these definitions will highly overlap. Hence, they collectively do not form a legitimate partition of their superset, which itself is in turn not a valid partition of the domain of knowledge.

Would you want to study with people who do not even get the very basic structure right of what they are teaching? I certainly do not recommend it, because at the highest level, i.e. the most visible structuring of what they do, they already get it completely wrong.

When you read a typical curriculum, you can only decide that the people who composed it, are utterly incompetent. Hence, you will learn nothing from them, except for becoming incompetent yourself.

But what is the thing signified? Why, that's a number! — Wayfarer

A number is an abstraction that always emerges in a Turing-complete axiomatic system. For example, in the core axiomatization of functions, i.e. the lambda calculus, numbers are functions:

There are several possible ways to define the natural numbers in lambda calculus, but by far the most common are the Church numerals, which can be defined as follows:

0 := λf.λx.x
1 := λf.λx.f x
2 := λf.λx.f (f x)
3 := λf.λx.f (f (f x))
...
Because the m-th composition of f composed with the n-th composition of f gives the m+n-th composition of f, addition can be defined as follows:

PLUS := λm.λn.λf.λx.m f (n f x)

Numbers are also set expressions in set theory, types in type theory, and combinator expressions in combinator calculus.

Whatever you pick as basic building brick for your Turing-complete axiomatization, you will always be able to express numbers as expressions in this brick.

Surprisingly, if you pick numbers themselves as the building brick, your axiomatization will not be Turing-complete. It will be much weaker.

Therefore, numbers are considered an uninteresting type of building brick and rather a byproduct of a better and more interesting building brick.

For example, set theory is much more powerful than number theory. In fact, you can rewrite number theory as some kind of byproduct inside set theory.

In other words, the idea that mathematics would be about quantities, i.e. numbers, is really wrong.

No, mathematics has quantitative relations as its subject matter — Dfpolis

Mathematics, science, and history are not subject matters.

They are epistemic domains, i.e. the sets of knowledge statements -- with knowledge a justified (true) belief (JtB) -- that you can legitimately justify using their associated epistemic justification methods.

There is no mathematical subject matter, nor a scientific subject matter, nor a historical subject matter.

If you can justify a claim using one of these methods, then the claim legitimately belongs to that epistemic domain. For example, any claim that you can justify using the scientific method, is part of the science epistemic domain.

Furthermore, these epistemic domains exclude each other. It is not possible that a proposition can be justified by one epistemic method and also by another.