But:
(I am: A Danish philosopher 63, currently engaged in writing a book adressing severe faults in modern Cosmology Science - Failures and misconceptions that - alas! - are made by illogical mathematicians with great ego'es. Research directed me to "Cut the not" for eventual answers. - Par example on: Lobachevsky and Bolyai built their geometries on the assumption C: through a point not on the line there exist more than 1 parallel to the line.)

YES
www.Cut-the-not is an interesting site. Thanks! Several things has been really good reading.

But - In your text - the "Manifesto" - you are - in my humble opinion - unfortunately making some false asumptions. This I know because I detest (or love/hate) math - It is so although I would surely want not to have these feelings about this interesting subject!

Not intended to be unfriendly, but:
As an example of your false assumtions you (I think) have written:
"The answer is simple. You cannot build an engine without good knowledge of Calculus." (Manifesto, §5)

That is simply just not true. I have done it as a child. Before I learned any calculus at school. It was a steam or rather air-pressured reaction-type toy-ship motor driven by a candle-light. The principle was:
Intake of water through a pipe below waterline with a little bullit valve. This pipe went into the heated metal case steam chamber, which then cooled and evaporated a little thus delivered portion of water. This rise in pressure created a blocking of the intake for a moment by forcing the little glass bullit to close the intake. Then while the generated steam was ejected through two pipes going backwards to the end of the boat, eventually, with not much pressure left, the process repeated its cycle of intake, evaporation, blocking and "jet exhaustion" and so on. Quite efficient - But of course not up-scalable, due to the principle, which needs fairly short distances for the steam to travel.

So - be sure: It is not the mathematic calculus that is the main ingredient in enginering and construction. It may of course though surely be a quite useful (and time-consuming) aiding tool. And it may even be indispensable when it comes to maintenance of any type of construction.
But it is the knowledge and experience and imaginative realisation and understanding of physical principles that matters most. And : Those insights you can have and acquire and use without any formal training and exercise involvoing "calculus".

But - Back to the core-subject:
You might wonder why I would have loved not to succumb to these love/ hate emotional barriers towards mathematics eventually?
Well - Because it would really have been wonderful if I had been able to do battle with sir Roger Penrose - (I suppose you know who he is) - on his own turf with no exceptions. He has advocated several not very well supported views in his book, titled "The Road to Reality" - And some beliefs of which he can be partially excused, as they date back to predecessors. - Beliefs that my research - I think - has identified as eventually being "shared common sense" amongst mathematicians.

However the contents of your site has convinced me that I shall probably never be able to feel comfort about mathematics. Especially not the "modern" part of it. (From about 1850 and forward.) It is simply too infested with ill conceived and illogical assumptions.

Lack of logic when it comes to the basic asuptions in Mathematics is what "blocks me out" and leave me behind its praised but - alas to me - unseen beauty.

Your site - as do several other sources - points once again towards Euclids Fifth Postulate as the stumble-stone for some - as I see it - common sense depleted, mathematical minds.

I simply don't get it - Obviously several of those fine people, who have som status of beeing master-mathematicians in the past centuries, have in vane tried to solve what they have seen as "the puzzle around The Fifth Postulate".
(Which to me is not a puzzle but really seems to be just a piece of cake to get around.)

As I percieve it an read the stories about these attemts, they have mainly been launched while trying to solve the Fifth Postulate by attempts to prove it as if it was a proposition - And mainly also by only using nothing but the previous four other postulates as axioms.

And also, as a neat contribution to the pancake of conceptual chaos which has been created these efforts, it has been a (to me) mindbugging praxis to nick-name The Fifth Postulate as the "Parallel Postulate". This has been done while ignoring the fact that no parallel lines at all are mentioned or pictured in the presentation of this postulate.

Gauss-Bolyai-Lobachevsky Space has been the strange outcome of these inapt efforts.

I nevertheles feel that I do miss something of the whole picture. I have researched for month, but without getting any closer to a satisfacorial solution. It is thus my modest hope that you might be kind enough to take your time to explain to me, what can be encompassed by two questions:

1) If we accept as a fact that "the Fifth Postulate" is not really a postulate, but a misplaced proposition, then what exactly is it that has persuaded the scolars to try to solve it while only using the previous four postulates - but - as I percieve it - without making (full) use of (all) the 23 previously proclamed definitions?
And:
2) Why has this history of inabilities then produced the wierd, self-invented rights i somebodys minds, which says it is o.k. to violate several of these 23 axiomatic definitions as it pleases, if you feel "you need to"?

I don't get it.

A definition is a definition in solo. You cannot have two contradicting sets of criteria for what - as an example - constitutes a straigt line.

But do I miss out on something in my quest for historical facts that can explain this breakdown of otherwise fully respected logic? That is what I am looking for.

Anyway - Thanks for your time -
Whether or not you chose to respond -

Bent K. Nielsen

Thank you for the kind words.

I hope you would not mind if I post you letter to the This and That forum at the CTK Exchange
(https://www.cut-the-knot.org/cgi-bin/dcforum/forumctk.cgi?az=list&forum=DCForumID4&conf=DCConfID1).

There may be more qualified people to answer your questions. (It's the answers you seek, do not you?)

Personally, I am uncertain as to which of the 23 Definitions does Gauss-Bolyai-Lobachevsky geometry violate.

Best,
A. Bogomolny

You can indeed have two definitions of what constitutes a straight line, but it leads to two different sets of definitions, one set for Euclidian geometry, and one set for Hyperbolic geometry. The latter ones definition of a straight line is not very satisfying if one is used to the Euclidian definition, since for example the total of the three angles of a triangle may not equal 180° degrees.

Likewise, you can look what happens if you drop any other of the postulates - it leads somewhere, but not to Euclidian geometry.

However, it is useful to use geometry different from Euclidian geometry to represent the theory of relativity.