What is Mathematics? - an addendum to the comments between Talbert and Maggot

GSTalbert1

Girlvinyl
Joined
Mar 31, 2012
Messages
6,754
Location
The Land of the Coon-Ass
As you can see, I don't know what are you tryng to say
and that is indeed my fault. I'm jumping around too much and I should be more considerate of the other person.

Tell me, in a few and simple words, what he's saying, pls.
Witt is not arguing that there is no wrong way of interpreting these mathematics, they didn't then (and we still don't) understand how exactly we are able to do mathematics. Wittgenstein is pointing out that in such a situation, while there are certainly wrong interpretations, because we lack the ability to fully understand what math is there arises the conundrum that two equally valid interpretations can arise in that neither can prove themselves or prove the other wrong. That does not mean they are useless, they are helpful in that they help us know what ends are dead and what ends are open. Good questions can only lead to better ones, not answers.


That's why HE made the reference to what HE would say to someone who used words incorrectly "Don't talk bosh" i.e. don't talk bullshit. This seems to draw upon the kantian tradition concerning the unknown, that such concepts while not providing the same epistemological validity as say a chemistry lab experiment or tasting a hot meal, we can none the less learn from the "shadows they cast upon the void."


*Sigh... Give me an example of an axiom and and example of a postulate

In modern maths, there's no difference between those terms.

In ancient Greece, there was a slight difference of scope, pretty subtle.
That would be a recent change, not a modern one. As you pointed out "Von Neumann" was very much interested in Axioms (don't know about Barnett) but that was because the guy was interested in fame and glory. From an early age he was interested in putting forth irrefutable axioms that would ground mathematics in axiomatic set theory in his doctoral thesis in 1925. Sadly his hopes for fame, glory, bitches and blow were dashed upon the rocks in 1930 by Kurt Godel's incompleteness theorem ended that. This seems to be something that happens repeatedly, von Neumann keeps missing the mark, as he did with game theory and John Nash <-- more on that further down. We've changed our viewpoint on mathematic axioms, which is why postulate now stands with axioms, but this is not something that existed from the start. To superimpose an anachronism on ancient mathematics and to not recognize the modern split with the Greek perspective is just silly.



And to say that obviety he needed years of lectures? What a Gaylord
No, he's working from Frege, who sent him to Russell with a personal recommendation (who was also working/continuing on Frege's work who was retiring). Russell's work (and Whitehead's) was developed into logical atomism. Wittgenstein's atomism was the split. Russell was trying to reduce everything into axiomatic parts, to get at the root of math and language. Witt argued that such atomism was impossible as you have music, were you cannot reduce a song to it's individual notes as the individual notes say nothing, or say a poem, where the meaning exists within the poem as a whole. There are complex aspects of communication that can be reduced to their component parts without destroying the meaning. The same is true of math. This is a rough picture of what he's talking about.

These lectures come after he broke with this perspective, during the 1940s.


WTF are you babbling about dude?
The split with Euclidian Geometry didn't fully set in until Newton was shaken from his pedestal. It takes time for the realization that axioms are useful bullshit, as I said, to set in. That doesn't change the fact that they are no more than useful bullshit ie axioms.
original assertion was: "Kim Peek never wrote a maths text. In fact, he couldn't explain the moral lesson of any child's book"
and my original assertion is that COMMUNICATION is not MATHEMATICS. Do you get me now sweetheart?

Not external/internal, that dicotomy is absurd. Reality has a mathematical structure, as demonstrated by the fact that huge chunks of it can be mathematically described with positive results.
You're claiming that the perfect number "2" or even "1" exists? Do tell!

So deaf people are worse matemathicians? Or are you saying that we have intuitive and innate grasp of multidimensional geometry, exotic manifolds and the cardinality of infinite sets?
How would a deaf person's with specific defective sensory organs deride the brain's ability to interpret sensory input?


No dude, we (except again, some geniuses) have intuitive grasp of thing like 2,+, 1/2, maybe square roots and negative numbers and the like. The most basic mathematical concepts. The 99,99999999999999% remaining it's not intuitive and comes from centuries of elaboration from axioms
LoL they knew about that before your "centuries of elaboration from axioms"

There, there...all is gonna be okay...who's the prettiest? who's the prettiest? Talby is!!!

Better?
That's what I'm talking about. We don't know how some of this shit works Maggot, what's more the human brain is simply remarkable. Even the functions we would call "basic" surpass our best technology. I don't believe in superior/inferior.

Please, explain in a few and simple words what in that sentence is deserving of a facepalm.
John Nash, one of those geniuses we are talking about, once marched into a fellow professor's office and drew two potato like blob on a chalk board and pointing at one stated, in horror, "This is heaven" then pointed at the other "this is hell." He also claimed he received secret messages "from outer space." When asked take "messages from outer space seriously?" Nash simply said that they had come to him the same way as his brilliant mathematical theorems. A genius isn't intrinsic, someone is only a genius in relation to something.


His point is that axioms, which are self-evident or universally recognized truths, are unnecessary in mathematics. That's about all I could gather from his text walls.
Not quite. Here's a thought experiment from Wittgenstein that is invaluable for this discussion.

I will have to paraphrase:

Imagine your with several other people and each person has a box, inside of which is something that everyone intends to refer to with the word "beetle". Further, suppose that no one can look inside another's box, and each claims to know what a "beetle" is only by examining their own box. Wittgenstein suggests that, in such a situation, the word "beetle" could not be the name of a thing, because supposing that each person has something completely different in their boxes (or nothing at all) does not change the meaning of the word; the beetle as a private object "drops out of consideration as irrelevant". Thus, Wittgenstein argues, if we can talk about something, then it is not private, in the sense considered. And, contrapositively, if we consider something to be indeed private, it follows that we cannot talk about it.


So how is it possible that anyone can talk about "beetle", or anything else for that matter considering that no one can "see" for themselves what "beetle" looks like for someone else?
 

CallMeMaggot

Girlvinyl
Joined
Jul 16, 2011
Messages
14,327
Better, but still far from...ergonomic, lol

that two equally valid interpretations can arise in that neither can prove themselves or prove the other wrong.

Valid interpretations of what? Of an ecuation? And wtf is an "interpretation" of a mathematical problem?

You mean, two different but valid results from a first grade ecuation, for example? Lol, no.

This seems to draw upon the kantian tradition concerning the unknown, that such concepts while not providing the same epistemological validity as say a chemistry lab experiment or tasting a hot meal, we can none the less learn from the "shadows they cast upon the void."

I ask you for simple and few words, and you can't fucking help to start sperging again at the first opportunity, haha

Ains...


That would be a recent change, not a modern one.

Vat

How subtle of a disctinction...

From an early age he was interested in putting forth irrefutable axioms that would ground mathematics in axiomatic set theory in his doctoral thesis in 1925. Sadly his hopes for fame, glory, bitches and blow were dashed upon the rocks in 1930 by Kurt Godel's incompleteness theorem ended that.

I think you are totally confusing here the impossibility of demonstrating the consistency of a system inside itself with the validity of its axioms

See:
1+1= 2 is an axiom. (kinda, but let's not get abtruse)
That you can't prove the consistency of the "+" function for the natural numbers inside the set of the natural numbers don't detract an iota of the validity of the "1+1=2" assertion, axiom, postulate or whatever you want to call it


To superimpose an anachronism on ancient mathematics and to not recognize the modern split with the Greek perspective is just silly.

True

Then, why did you do that, man?

It's you who came with the ancient Greece definition of axiom, which nobody uses today...and now you cry about anachronisms?

Dude, I really don't know sometimes if you are pulling my leg, or are just fucking nuts



1º you cannot reduce a song to it's individual notes as the individual notes say nothing, or say a poem, where the meaning exists within the poem as a whole. 2º There are complex aspects of communication that can be reduced to their component parts without destroying the meaning. The same is true of math. This is a rough picture of what he's talking about.

Is it me, or sentence 1 is almost the exact opossite of sentence 2 ?

You say opossite things in the same paragrph without blinking.

And then get surprised that nobody gets what you are saying, lol


The split with Euclidian Geometry didn't fully set in until Newton was shaken from his pedestal. It takes time for the realization that axioms are useful bullshit, as I said, to set in. That doesn't change the fact that they are no more than useful bullshit ie axioms.

You original sentence was "...pre-quantum axiomatic model of mathematics bla bla...", and that was the source of my question "WTF are you babbling about dude?"

What euclidian geometry has to do with that, beats me.

In fact, you totally ignore the source of my question (which was wtf has to do quantum mechanics with mathematical axioms, modern or ancient) and take the oportunity to rant about something utterly unrelated , and to repeat that axioms are bullshit...

YOLO# I guess...




and my original assertion is that COMMUNICATION is not MATHEMATICS. Do you get me now sweetheart?

Not in the slightless, honeybum

Let's recapitulate the exchange:

1º I said: "Kim Peek never wrote a maths text. In fact, he couldn't explain the moral lesson of any child's book"

2º You answered: :picard:

3º I answered to that with: "Please, explain that facepalm"

4º Your final answer to my request is: " my original assertion is that COMMUNICATION is not MATHEMATICS."

TEN POINTS FOR CLARITY AND COHERENCE, MOTHERFUCKER

clap, clap, clap

Dude, I'm really feeling right now that communicating with you in a sensible manner is impossible :(



You're claiming that the perfect number "2" or even "1" exists? Do tell!

Yes, I guess. They certainly exists, and in pure perfection, inside the mind of any guy who has to do this operation: "1+2"

And your point is...?

How would a deaf person's with specific defective sensory organs deride the brain's ability to interpret sensory input?

Wat

I dunno, tell me. And I don't know either how that has anything to do with my question, which was "Are you talking about option A or B"?

As a side note...do you realize how I almost always answer your direct questions...and how you almost never do so, choosing to ask another (usually unrelated or obscure) question instead?

Another 10 points for clarity and fair game :rimshot:


LoL they knew about that before your "centuries of elaboration from axioms"

Zenon paradoxes has NOTHING to do with "multidimensional geometry" and "exotic manifolds" and only really tangentially with "the cardinality of infinite sets", which were the 3 examples I put on the table.

Also, paradoxes, as its name imply, are puzzling... and not a tool for calculus, at all.

Multidimensional mathematics, for example, on the contrary tries to take the puzzlement out of the subject, and ARE a tool for calculating things. A tool that works, btw.

So your pretension that some ancient greeks knew about, let's say, Fourier transform because they found some puzzling broad logical paradoxes is ...is...well, I'll not get too florid: it's utterly stupid

That's what I'm talking about.

About you being the prettiest, which was my remark at that point?

LOL, at last some sincerity


We don't know how some of this shit works Maggot, what's more the human brain is simply remarkable. Even the functions we would call "basic" surpass our best technology.

True, true...and totally irrelevant to this discussion, axioms included, far as I can see


John Nash, one of those geniuses we are talking about, once marched into a fellow professor's office and drew two potato like blob on a chalk board and pointing at one stated, in horror, "This is heaven" then pointed at the other "this is hell." He also claimed he received secret messages "from outer space." When asked take "messages from outer space seriously?" Nash simply said that they had come to him the same way as his brilliant mathematical theorems. A genius isn't intrinsic, someone is only a genius in relation to something.

HAHAHA

I asked: "Please, explain in a few and simple words what in that sentence is deserving of a facepalm."

---and you come with a fucking abstruse parable?

Srly?

You ARE definitely trying to pull my leg. Or my dick. Aren't u? Confess!!

:D:D:D


Sweet Jesus...
 

GSTalbert1

Girlvinyl
Joined
Mar 31, 2012
Messages
6,754
Location
The Land of the Coon-Ass
that two equally valid interpretations can arise in that neither can prove themselves or prove the other wrong.

Valid interpretations of what? Of an ecuation? And wtf is an "interpretation" of a mathematical problem?

You mean, two different but valid results from a first grade ecuation, for example? Lol, no.
No, he says specifically he leaves everything as it is. There are different interpretations of what exactly math is, that it is not referring to just how we do it or what it does but what actually is it. We don't really have much understanding of exactly how our brains work. We can speculate, and in those speculations there may be the kernel of truth but unless it is a perfect or complete understanding there exist imperfections, blind spots where we do not know. Thus you can have two propositions that are, relatively, equally right and equally wrong and thus strike at an issue with the current state of human knowledge for currently it is always finite, therefore it is possible to survey it. Thus a good question should really just lead to better ones.

This seems to draw upon the kantian tradition concerning the unknown, that such concepts while not providing the same epistemological validity as say a chemistry lab experiment or tasting a hot meal, we can none the less learn from the "shadows they cast upon the void."

I ask you for simple and few words, and you can't fucking help to start sperging again at the first opportunity, haha

Ains...
Say two people each give propositions that appear equally valid yet are contradictory. After fully examining their foundations, you can find no flaw or contradiction in their reasoning then you know that you all are but stumbling in the dark, i.e. "casting shadows on the void." That is what is meant when Kant says he "made space for faith by doubting knowledge."
 

CallMeMaggot

Girlvinyl
Joined
Jul 16, 2011
Messages
14,327
Say two people each give propositions that appear equally valid yet are contradictory. After fully examining their foundations, you can find no flaw or contradiction in their reasoning then you know that you all are but stumbling in the dark, i.e. "casting shadows on the void." That is what is meant when Kant says he "made space for faith by doubting knowledge."
That's why I stopped reading about conspiracy theories, too disturbuing :D

Nothing to do with maths, tho
 

GSTalbert1

Girlvinyl
Joined
Mar 31, 2012
Messages
6,754
Location
The Land of the Coon-Ass
Nothing to do with maths, tho
Yes it does, no one is saying that mathematics can't be done until we understand it, for that would be absurd as saying we must delay digestion till we had finished the study of anatomy and physiology. It is a proposition to view mathematics as an object of a science field. Improvements in a field are essentially improvement of the field as a whole. This is not a statement of theory but definably a matter of historical fact.

Turing: I understand but I don't agree that it's simply a question of giving new meanings to words.

Witt: Turing doesn't object to anything I say. He agrees with every word. He objects to the idea he thinks underlies it. He thinks we're undermining mathematics, introducing Bolshevism into mathematics. But not at all.

Valid interpretations of what? Of an ecuation? And wtf is an "interpretation" of a mathematical problem?


You mean, two different but valid results from a first grade ecuation, for example? Lol, no.

I think you are totally confusing here the impossibility of demonstrating the consistency of a system inside itself with the validity of its axioms
Not quite an interpretation in that sense because then it wouldn't be an interpretation of valid results of a mathematical problem.

Turing: You cannot be confident about applying your calculus until you know that there are no hidden contradictions in it.

Witt: There seems to me to be an enormous mistake there. For your calculus gives certain results, and you want the bridge not to break down. I'd say things can go wrong in only two ways: either the bridge breaks down or you have made a mistake in your calculation - for example you multiplied wrongly. But you seem to think there may be a third thing wrong: the calculus is wrong.

Turing: No. What I object to is the bridge falling down.

Witt: But how do you know that it will fall down?

Turing: If one takes Frege's symbolism and gives someone the technique of multiplying in it, then by Russell Paradox he could get a wrong multiplication.

Witt: This would come to doing something which we would not call multiplication.
1+1= 2 is an axiom. (kinda, but let's not get abtruse)
That you can't prove the consistency of the "+" function for the natural numbers inside the set of the natural numbers don't detract an iota of the validity of the "1+1=2" assertion, axiom, postulate or whatever you want to call it
Exactly, we have the experience of 1+1=2. To have 1+1=3 would be referring to a completely different experience. These are just representations for our common experience of mathematics, but they are not mathematics proper. As to the problem of axioms, I am referring to the Euclidian systems that Frege's Symbolism represented. When I spoke of the quantum revolution I was referring to papers published during the Year of Miracles. It is not mere conjecture that the human understanding of mathematics and it's relationship with the world has greatly improved, but many of mathematics' dead ends run rampant throughout Academics as vestigial peculiarities; they are much akin to a bunch of chickens running round with their heads cut off and I know we both agree on that.

Look at this paper on Consumption and Identity -

What of the traumatic loss of self-signifying posessions before death? Those who have lost self-signifying possessions to theft often report feelings of violation and pollution akin to being raped (Maguire 1980; Papp 1981; Van den Bogaard and Wiegman 1991).
This is an example of a bunch of scientists discovering a new way to say "Loosing things can be bad, perhaps even traumatic."
 

GSTalbert1

Girlvinyl
Joined
Mar 31, 2012
Messages
6,754
Location
The Land of the Coon-Ass
Socrates: motherfucking cheater, lmao. Of course he was directing that child to give the correct answers. Try that with the one raised in the privation tank, and see if "the truth" is in him too-lol
Witt agrees.

"Reading the Socratic dialogues one has the feeling: what a frightful waste of time!" -Wittgenstein

An important thing to note is that there's a passage where Socrates makes exactly the same point you are, meaning he was aware of his cheating.

"The Philosopher teaches the truth not by inventing parables (or by using contradictions between parabolic statements), but by using conscious and intentional contradictions, hidden from the vulgar, between unparabolic and unenigmatic statements." - Leo Strauss

Mind, these quotes compliment your intelligence in no small way.
 

Aroukar

EDF Elite
Joined
Mar 12, 2012
Messages
4,055
Location
America
I have read barely a couple of philoshopy books in my life, not too savy on it...but I consider myself more or less proficient on detecting bullshit :D

That's a good balance. I can't take someone (who relies too much on theory) seriously.
 

GSTalbert1

Girlvinyl
Joined
Mar 31, 2012
Messages
6,754
Location
The Land of the Coon-Ass
Two fun facts about Witty -
  1. As a young engineering student he patented a novel aircraft engine that anticipated the jet engine and was tested and reinvented in 1943.
  2. At a Social Democratic Workers demonstration in Vienna the police opened fire killing 85 demonstrators and bystanders. Witty was with his sis Maragarete who tried to turn tail and flee he, a WWI decorated vet, he kept his cool & coldly scolded "When one hears rifle fire, one doesn't run."
This is what happens when you inject philosophy into science @MrGask
Well, philosophy is the science, or the wisdom, of questioning. Science without philosophy is questioning in ignorance. A better example would be Heinrich Hertz's Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt (Principles of Mechanics Presented in a New Form) where he applies philosophy as it is meant to be applied, therapeutically, and solves the riddle of "what is force" by simply presenting a reconstruction of the Newtonian system but without using 'force' as an axiomatic concept:


Hertz said:
When these painful contradictions are removed, the question as to the nature of force will not have been answered; but our minds, no longer vexed, will cease to ask illegitimate questions.
The Prinzipien der Mechanik was one of the books Witt read as a teenager and this pretty much sums up the essence of what philosophy is meant to be.

This was not idle chatter, there is an excellent example of his aid researching treatment of victims of air raid casualties suffering from "wound shock" victims during WW2 in England.

Drs Roland Grant had already reported great frustration in his attempts at studying this phenomenon:
Recent experience of air raid casualties shows that in spite of all the work already done, specially in the last war, but little is known about the nature and treatment of traumatic or wound shock. In the first place there is in practice a wide variation in the application of the diagnosis of 'shock'. We cannot yet foretell and we are often in doubt about treatment. Moreover, the lack of a common basis of diagnosis renders it impossible to assess the efficacy of the various methods of treatment adopted.

There is good ground, therefore, for the view that it would be better to avoid the diagnosis of 'shock' and to replace it by an accurate and complete record of a patient's state and progress together with the treatment given.
The reception from the medical establishment ranged from lukewarm to chilly. From the response of Colonel Whitby of the Army Blood Transfusion Service sent to the Medical Research Council on Dr. Grant's report:


Quite a lot of the preamble, and some of the discussion was devoted to a diatribe against the word 'Shock'. I do not feel that this point needs to be so heavily emphasized.

It is not justifiable to throw over the findings of the last war. These men were not fools, they at least established the fundamental fact that lowered blood pressure was a sign very constantly observed. Grant would throw over the whole of the valuable MRC literature of the last war because their records do not attain his standard of detail.
Witt contacted the two doctors and they enthusiastically took him in as part of their research team.
Dr. Grant in a letter to a colleague: Proffesor Ludwig Wittgenstein, of whom I told you, joined the unit as Laboratory Assistant. He is proving very useful.

Both of the doctors recall Witt's influence which played an invaluable role in not only their thinking, but also the style and structure of their final report Observations on the General Effects of Injury in Man. A selection:

In practice we found that the diagnosis of shock seemed to depend on the personal views of the individual making it rather than on the generally accepted criteria. Unless we were acquainted with these views we did not know what to expect when called to the bedside. The label alone did not indicate what signs and symptoms the patient displayed, how ill he was or what treatment he required. The only common ground for diagnosis that we could detect was that the patient seemed ill. We were led, therefore, to discard the word 'shock' in its varying definitions. We have not since found it to be of any value in the study of injury; it has rather been a hindrance to unbiased observation and a cause of misunderstanding.
This report was far better received and had precisely the effect Witt was gunning towards: ending misguided lines of research. A quote from the Report of the Medical Research Council for 1939-45 on Grant's research:

(Grant's report) threw grave doubt upon the value of attacking the 'shock' problem as if wound 'shock' were a single clinical and pathological entity. In consequence, several lines of investigation started for the Committee at the beginning of the war were abandoned.

Some relevant quotes from Witt on the subject
  • I am sitting with a philosopher in the garden; he says again and again 'I know that that's a tree', pointing to a tree that is near us. Someone else arrives and hears this, and I tell them: 'This fellow isn't insane. We are only doing philosophy.
    • Leo Strauss makes a similar remark: Only a great fool would call the new political science diabolic: it has no attributes peculiar to fallen angels. It is not even Machiavellian, for Machiavelli's teaching was graceful, subtle, and colorful. Nor is it Neronian. Nevertheless one may say of it that it fiddles while Rome burns. It is excused by two facts: it does not know that it fiddles, and it does not know that Rome burns.
  • “What we find out in philosophy is trivial; it does not teach us new facts, only science does that. But the proper synopsis of these trivialities is enormously difficult, and has immense importance. Philosophy is in fact the synopsis of trivialities.”
  • The following is a question I constantly discuss with Moore: Can only logical analysis explain what we mean by the propositions of ordinary language? Moore is inclined to think so. Are people therefore ignorant of what they mean when they say ‘Today the sky is clearer than yesterday’? Do we have to wait for logical analysis here? What a hellish idea! I must, of course, be able to understand a proposition without knowing its analysis.
    • Hegel has similar commentary: The neglect of this distinction between thought in general and the reflective thought of philosophy has also led to another and more frequent misunderstanding. Reflection of this kind has been often maintained to be the condition, or even the only way, of attaining a consciousness and certitude of the Eternal and True. Such a doctrine would find its parallel, if we said that eating was impossible before we had acquired a knowledge of the chemical, botanical, and zoological characters of our food; and that we must delay digestion till we had finished the study of anatomy and physiology. Were it so, these sciences in their field, like philosophy in its, would gain greatly in point of utility; in fact, their utility would rise to the height of absolute and universal indispensableness. Or rather, instead of being indispensable, they would not exist at all.
  • It is all one to me whether or not the typical western scientist understands or appreciates my work, since he will not in any case understand the spirit in which I write. Our civilization is characterized by the word ‘progress’. Progress is its form rather than making progress one of its features. Typically it constructs. It is occupied with building an ever more complicated structure. And even clarity is sought only as a means to this end, not as an end in itself. For me on the contrary clarity, perspicuity are valuable in themselves.

    I am not interested in constructing a building, so much as in having the perspicuous view of the foundations of possible buildings.

    So I am not aiming at the same target as the scientists and my way of thinking is different than theirs.

  • We never arrive at fundamental propositions in the course of our investigation; we get to the boundary of language which stops us from asking further questions. We don’t get to the bottom of things, but reach a point where we can go no further, where we cannot ask further questions
 

uberfukken

Custom title
Joined
Nov 22, 2011
Messages
23,560
Personal attacks from people who can't/refuse to understand it?
70% of philosophy is drawing psychological comparisons and the other 30% is phrasing it in a way that makes you sound smart. Math is an observational tool and craft. Might as well examine the foundations between Aristotle and a hammer.
 

GSTalbert1

Girlvinyl
Joined
Mar 31, 2012
Messages
6,754
Location
The Land of the Coon-Ass
70% of philosophy is drawing psychological comparisons and the other 30% is phrasing it in a way that makes you sound smart. Math is an observational tool and craft. Might as well examine the foundations between Aristotle and a hammer.
Wittgenstein:

Philosophy consists of logic & metaphysics: logic is its basis.

Epistemology is the philosophy of psychology.

Distrust of grammar is the first requisite for philosophizing.

Propositions can never be indefinables, for they are always complex.
In philosophy there are no deductions: it is purely descriptive.

Philosophy gives no pictures of reality.

Philosophy can neither confirm nor confute scientific investigation.

Philosophy is the doctrine of the logical form of scientific
propositions (not only of primitive propositions).

The word “philosophy” ought always to designate
something over or under, but not beside, the natural sciences.
 
Top