Well, I still don’t know what you mean by irrational. That must be solved first.
I think there is a confusion between the maths’ “irrational” and whatever you mean by “irrational”.
In maths, as Magnus has pointed out, an irrational number is merely a number that cannot be expressed in a certain way. There is nothing illogical about the number itself. It is merely an issue of how it can be expressed - it must be able to be expressed by a ratio of integers - “non-ratio” - it cannot be rationed out into decimals. Why they do that - I don’t know. Why they care - I don’t know. There are numbers that cannot be expressed in decimal form (such as pi). Still there is nothing irrational about pi except that it cannot be expressed in decimals. The resolution is to truncate it - use an estimate accurate enough for the end need.
But I don’t think you are talking about ideas that cannot be expressed in a particular language. You seem to be talking about ideas that are incomprehensible or contrary to logical structure - perhaps oxymorons like a square-circle. That is a different issue than maths concerns. Math has no numbers that are illogical. Math is just logic applied to quantities - all math is just logic applied to quantities. ALL math is logical.
So as I said before, we need an example of what it is that you mean when you speak of the “irrational”. I don’t think an irrational number is what you really mean - because those have nothing to do with philosophical issues (certainly nothing to do with political issues).
There is no confusion, because I have spoken about the suggestion that one should start from the mathematical definition (read my text about it), in order to come afterwards to a philosophy of the irrational.
I never used the definition of the mathematical “irrational”, i.e. the one for the “irrational numbers”, together with the linguistic (lexical) definition or the usage in the common language.
@ Magnus Anderson.
The irrational is a cognition, thought, speech or action without the participation of the rational. More precisely, it is something characterized by no or insufficient use of reason and by a transcendent (not transzendental!) use.
@ Others.
I agree. The irrational is the unreasonable (illogical), is not or not completely comprehensible by the ratio, is not accessible to the logical thinking, is that which the rational simply or yet almost completely lacks.
So we are just talking about beliefs that have no reasoning or rationale to back them up.
I have not read everything here. But I think it’s up to you whether you want to talk about belief or something else. The title of this thread is: “Can philosophy integrate the irrational as mathematics can”. So this thread is not only about irrationality, which means that you can just as well talk about integrating the irrational into the rational, and that is exactly what has happened here so far, at least judging by what I have read. According to the title of this thread, I would say that it is primarily about the integration of the irrational into the rational through philosophy, and this integration can be compared to the one that mathematics has done with the irrational numbers.
That is what I originally thought - but –
who knows. He won’t give us an example except in maths.
And there is more if read between the finer lines
Great again wrote:
“The irrational is a cognition, thought, speech or action without the participation of the rational. More precisely, it is something characterized by no or insufficient use of reason and by a transcendent (not transzendental!) use.”
The participation ( or integration may refer to both the irrational and rational, and here is the important focus: as related both sets, except it’s not made clear weather it considers a logical nexus, or a semantic one, and weather they intersect, creating a subset, or an evolving objective synthesis. This is why the comments do not appear to have neither a common theme nor degrees of separation or union.
As it stands, the partial element he refers to, the ‘forest’ may suggest a mystical participation.( Levy Bruhl)
That was the comment that made me think he must be talking about merely non-reasoned beliefs - opinions.
And especially this:
“More precisely, it is something characterized by no or insufficient use of reason and by a transcendent (not transzendental!) use.”
I think he appears to neglect the and -Or. ; either -or distinction.
Why don’t you just stick to the title of this thread?
Besides, the relationship of the rational and irrational numbers as subsets of the set of real numbers has long been pointed out. I guess you didn’t read that.
Moreover, the title of the thread pretends that the point is not to integrate the rational into the irrational, but to integrate the irrational into the rational. Just read it.
@ Obsvr, Meno.
Magnus Anderson asked for a definition for the “irrational”. That is why I gave him this definition ( ilovephilosophy.com/viewtop … 3#p2816886 ). So it is just a definition for the “irrational”. Wikipedia gives almost the same definition:
However, this thread is not only about the irrational, as Sleyor Wellhuxwell also pointed out, but it is mainly about how philosophy can manage to integrate the irrational into the rational. As an example I mentioned mathematics, which can be a pattern for integration, but of course is not in a 1:1 relation to philosophy. However, all this can already be read in the Opening post and in the further course in many of my posts.
I have also given several examples. Only you don’t seem to accept them.There is no rational explanation for why Ludolf’s number exists or for the fact that this number corresponds exactly to the amount/value to which it corresponds.There is no rational explanation for why Ludolf’s number exists or for the fact that this number corresponds exactly to the amount/value to which it corresponds. There is also no rational explanation for why there is gravitation, if there is gravitation. This and more why-questions I have posted. Also the love I have mentioned as an example there ( ilovephilosophy.com/viewtop … 5#p2816838 ). There are innumerable examples. There are more examples for irrationality than for rationality. Irrationality does not mean simply nonsense. Explain to me once, Meno or Obsvr, why the universe exists, if it exists, or, especially for you, Obsvr, why there is affectance, if it exists. In the end, one ends up in almost all cases with God or with the question why in the universe (the “nature”) everything is set the way it is set (compare “constants”) or with the question: Why is there being and not rather nothing? One cannot give rational answers to these questions. Therefore, one should consider the irrational. If one never tries it, then one also cannot know whether the irrational can help us to understand everything in all better (thus: rationally, as the mathematicians do).
Can we say that it is something like a philosophical experiment or a metaphysical experiment?
If you don’t mean it satirically: yes.
Just look at this globalistic chaos called “Corona”, in which the irrational seems to have won. Seriously: The irrational dominates more anyway. You are lucky if you live on an island called “the West”, because the rational had been ruling there for a long time. This now seems to be coming to an end.
But we cannot fight this irrational simply with rational. That is not enough for reasons that I have already addressed here almost countless times. The conclusion is that we must study the irrational, insofar as we can, and try to integrate it, which means that we must try to make it serve us. The rational has only this one possibility if it wants to stand against the irrational, which is on the rise everywhere.
I know what you mean. We are facing a planned dumbing down - the irrational is globally on top anyway - books are no longer read, only the nonsense on the internet, which is contaminated more and more with irrational stuff. And in addition to that, the culture in which the rational has been overdimensionally strong now sees itself exposed to an irrational power and does not know how to defend itself against it, especially since demography also contributes to the fact that this process runs exponentially.
Rage is also a good example of irrationality. But dealing with rage, again, can be something rational, and it should be.
In the West, however, rage is also a taboo. For it the eros is all the stronger: the greed!
Some will now ask for definitions again.
“Define the word ‘again’.” 
Although not an accurate statement, That is the kind of example we have been asking for.
The “examples” that you have been giving have been within maths. We understand those without examples -
We are familiar with those kind of numbers. What we were asking for was something specific outside of maths that would clarify to us what irrational things that you would like to see integrated. You have finally supplied us with one example - “gravitation”.
Obviously you have not read Mithus’ book or James’ posts on those every issues. James went to extreme detail as to exactly what logic leads to the necessary existence of all that exists - including gravitation - why it does what it does - exactly how it works - exactly why it even exists (even why space exists). He was extremely into the whys for everything. But he was also an extreme rationalist - to the point of giving rational definitions for “God” (many of them - although not depending on those in his explanation of “natural phenomenon”).
I’ll now dig up some of James’ posts from this board (also revealed in Mithus’ book) to show you how James "integrated the [seemingly] irrational [or unexplained] into rational thought concerning gravitation - I think he even made a thread on that one — give me a few moments —
I suspect the word “unexplained” would have circumvented a lot of the confusion.
That is another example that James explains with his Physics of Psychology (that I am currently studying and also in her book).