Continued from previous
- jamma246
- The modern approach has proven useless? I'm going to assume that this was a joke. For practical as well as philosophical application, the modern approach in unparalleled and subsumes those naive philosophies of which you speak.
Just because the methods in modern mathematics aren't instructive to you, that doesn't mean that they aren't instructive for others. The examples are endless. To take one, Lawvere found that the theory of metric spaces became much more elegant after allowing distances to become infinite. To someone like you, that wouldn't be 'allowed', since infinity doesn't 'exist'. You would never have progressed mathematics or logical thought like he did, as a result of your dyed-in-the-wool philosophies. But he didn't think in this way, thankfully. And there is nothing unintuitive about any of this: if two points are infinite distance from each other, then that is like one point just being unreachable from the other (in a sense, they are on distinct 'islands' of points). Metric spaces don't necessarily need to represent physical objects, they could represent theoretical objects too, so points at infinite distance could, for example, represent points where one can't be reached through a finite process to the other, such as a calculation.
Another example is given by the whole field of analysis. It is all intuitive, all elegant, all instructive. But it is basically essential to use the concept of infinity, in particular for sequences. The notion of a 'limit' is what took the field off the ground, this allowed Newton and Leibniz to make the great progress that they did, in mathematics as well as philosophy. Infinite sequences are something that you seem to take exception to in the video.
- Hans-Georg Lundahl
- "The modern approach has proven useless? I'm going to assume that this was a joke."
Preliminarily, I hope?
Because, it was not.
For practical uses the modern approach to theory is useless since no practical needs of any value require infinite decimals or infinite continuousity of a fraction to be actually executed.
For theoretical uses, YOU just proved the modern approach useless, since it made you incapable of answering points that were just silly, while as a Thomist I could easily answer them.
"For practical as well as philosophical application, the modern approach in unparalleled and subsumes those naive philosophies of which you speak."
Being unparalleled for application is no guarantee of being very useful. Philosophical applications can namely be very wrong, like Kant and Krauss have shown. Practical applications that are provenly right are also, systematically something else than really applying the modern approach.
"To take one, Lawvere found that the theory of metric spaces became much more elegant after allowing distances to become infinite."
To St Thomas and to me, he is quite welcome to the elegance, and even of using infinity as a model (and never actually using it), as long as he doesn't take that for factual truth about infnite distances really existing. Btw, I didn't say infnity doesn't exist. I said that infinite magnitudes and multitudes do not exist.
"You would never have progressed mathematics or logical thought like he did, as a result of your dyed-in-the-wool philosophies."
You missed that St Thomas gave mathematicians quite enough leeway to do anything they wanted with fictional concepts in maths.
Two things I balk, one is taking "infinite distance" as a non-fiction mathematical reality, or "number line" or "numbers less than zero"; the other is taking solid concepts like finite numbers (and their always being rational) or like geometric figures (and their size ratios not always being rational) and call them fictions just because fictions exist in maths.
"And there is nothing unintuitive about any of this: if two points are infinite distance from each other, then that is like one point just being unreachable from the other (in a sense, they are on distinct 'islands' of points)."
In that sense there can be no infinite DISTANCE between them because a point in Archenland or Narnia or Elidor (supppsing God had created such worlds) has no spatial relation to our world at all.
A distance between the points implies a line reaching one point from the other point, and that means it is finite. As you just very correctly said "infinita non est pertransire" (as St Thomas applied in the Cosmological proof of God), where the distance or rather non-nearness of two points is infinite, it is no distance and where it is a distance it is not infinite.
"Another example is given by the whole field of analysis. It is all intuitive, all elegant, all instructive. But it is basically essential to use the concept of infinity, in particular for sequences. The notion of a 'limit' is what took the field off the ground, this allowed Newton and Leibniz to make the great progress that they did, in mathematics as well as philosophy. Infinite sequences are something that you seem to take exception to in the video."
Analysis is an art of approximation, thus of fiction, though useful such.
A limit implies a sentence like "if the sequence could go on to infinity, even so the result would not progress beyond, nor regress behind this limit."
I agree, especially about the use of the unreal mood. The value you get for a limit may be theoretically equivalent to the series going on to infinity, but it can never actually be had that way. The limit value is, essentially, one thing. The series is another and only by the impossible operation of making it go on to infinity (as opposed to St Thomas Aquinas' "as long as needed", check the quote above again) could one thinkably adjust it to the value called limit value without any use of approximation.
In other words, the series is and will always remain as distinct from the limit value as a series like:
3:1 / 4:1
31:10 / 32:10
314:100 / 315:100
3141:1000 / 3142:1000
31,415:10,000 / 31,416:10,000
will always fall on one or other side of the real value of π.
- jamma246
- "For theoretical uses, YOU just proved the modern approach useless, since it made you incapable of answering points that were just silly, while as a Thomist I could easily answer them."
Name some of those points, I'd like to hear them.
- Hans-Georg Lundahl
- The ones YOU named. Are you forgetful?
Let me help you out :
[Linking to this message]
- jamma246
- "A limit implies a sentence like "if the sequence could go on to infinity, even so the result would not progress beyond, nor regress behind this limit.""
This just proves that you don't understand how infinity is rigorously used in mathematics. This interpretation of what a limit is is completely incorrect.
There is no appeal to "if the sequence could go on to infinity". The sequence is an infinite string. And there is no issue with that whatsoever, no problems occur. And the definition of a limit of such a thing has a simple and well-defined definition.
If you would just see how real mathematics is done, I think that you would find that it is always logical, and there is usually an urge to keep things conceptual and intuitive. Often, using some notion of infinity is incredibly helpful and philosophically the 'correct' thing to do. You seem to take the position that, fine, this is useful as a trick, but has no application to the real world. But you have no justified reason for believing this. And mathematics, at heart, is simply a bag of tricks. It is the most "real" of the sciences, in that it is all based on purely logical reasoning, so its study is always applicable in that sense. But trying to partition the "real and non-real", I take exception to that. It is firstly very arrogant, since it implies that the human can know the distinction. Secondly, I think that it is a meaningless distinction. All concepts of mathematics are "non-real" in the sense that they are abstractions. So your bleating about how this and that are "non-real" or "don't exist" is not only arrogant, but also meaningless.
Please - I honestly and sincerely mean this - read up on some modern theory. Read a book about metric spaces or analysis or set theory. It will open your eyes. Mathematicians are completely aware of these philosophical issues. The ideas of constructivism are still thought about in areas such as topos theory. The axiom of choice is still an important and recognised issue. Read, and open your eyes - get out of the 13th century.
- Hans-Georg Lundahl
- "The sequence is an infinite string."
Only potentially so.
"And there is no issue with that whatsoever, no problems occur. And the definition of a limit of such a thing has a simple and well-defined definition."
Let us take an example of limes and sequence, shall we.
Fibonacci sequence has for limes φ, right?
1:1 / 1:2
2:3 / 3:5
5:8 / 8:13
13:21 / 21:34
However long the sequence may go on in the operation of the mathematician, it will certainly converge with but never reach the limes φ.
Replacing the finite values of sequence with infinity-of-that-sequence is really stepping out of the sequence.
We don't need that, we don't need an arithmetic definition of φ to be rigourous, we already have exact geometric definitions.
"If you would just see how real mathematics is done, I think that you would find that it is always logical, and there is usually an urge to keep things conceptual and intuitive."
Oh, mathematics as done in the 13th C. is no longer real mathematics? Since when?
"Often, using some notion of infinity is incredibly helpful and philosophically the 'correct' thing to do."
And philosophically the incorrect thing to do, vide Sanctum Thomam.
[OK, exaggerated, since "some notion of the infinite" covers also the purely potential one, which is quite acceptable to St Thomas, but I got a bit flustered.]
"You seem to take the position that, fine, this is useful as a trick, but has no application to the real world. But you have no justified reason for believing this."
My justified reason is observation of mathematicians at every turn of their real practise.
In the sequence of fractions that man did not go on into infinity, he stopped at a very early level of execution of the series.
"And mathematics, at heart, is simply a bag of tricks."
Algebra may have given you that impression. Confusing arithmetic and geometry may also have given you that impression. But get INTO the thirteenth century, see them reason on maths or take me as a default, and we shall see if statements about maths can't be made to fit real mathematical realities of an obvious nature a good deal better.
"But trying to partition the "real and non-real", I take exception to that. It is firstly very arrogant, since it implies that the human can know the distinction."
Oh, my condoleances! You are a damned Kantian. Damned unless you repent that is.
Yes, one can know the distinction quite well, and your saying one can't is ANOTHER illustration on your part that the modern take is hurting your grasp on the philosophy of maths.
For instance, the squares of binomials. (a+b)^{2} = a^{2} + 2ab + b^{2} both per se and as a mathematical convenience.
But (a-b)^{2} = a^{2} - 2ab + b^{2} only as a convenience and per accidens. Per se it rather equals (the more cumbrous, I admit) a^{2} - ab - (ab - b^{2}).
"Secondly, I think that it is a meaningless distinction. All concepts of mathematics are "non-real" in the sense that they are abstractions."
That is misunderstanding what abstracting means.
Abstracting is not and should not be inventing mere placeholders. Three is indeed an abstraction, but of real (and imagined but realistically so) collections of three objects or of three events etc. Square is an abstraction of real (or imagined but realistically so) square formed surfaces. All through the real universe. Inventing counters and abstracting concepts correctly are two different things.
"So your bleating about how this and that are "non-real" or "don't exist" is not only arrogant, but also meaningless."
To your Gaussian barbarism from 19th C. Germany, no doubt.
"Read, and open your eyes - get out of the 13th century."
I have read and opened my eyes and landed IN the 13th century as far as mental furniture is concerned.
Thank God!
- jamma246
- "Only potentially so."
Meaningless.
"Let us take an example of limes and sequence, shall we."
What is a lime?
"However long the sequence may go on in the operation of the mathematician, it will certainly converge with but never reach the limes φ.
Replacing the finite values of sequence with infinity-of-that-sequence is really stepping out of the sequence."
Who says that it will "reach" φ? What is "infinity-of-that-sequence"? What is "stepping out of the sequence"? It's like you are inventing things that no mathematician does to justify your stupid position.
"My justified reason is observation of mathematicians at every turn of their real practise."
Well then, you seem to have little experience of working with mathematicians.
"In the sequence of fractions that man did not go on into infinity, he stopped at a very early level of execution of the series."
What do you mean by "execution of the series"? You are talking meaningless, non-rigorous babble.
"...and we shall see if statements about maths can't be made to fit real mathematical realities..."
Exactly. Mathematics is always made to "fit" reality when one wishes to apply it, it isn't actually reality. It is abstraction. So your distinction is meaningless. All mathematics is abstraction, it's just that some concepts have simpler connections to reality than others.
"But (a-b)sqrd = a^2 - 2ab + b^2 only as a convenience and per accidens. Per se it rather equals (the more cumbrous, I admit) a^2 - ab - (ab - b^2)."
Under what rules? This is a preposterous sentence, utterly hilarious.
Continue enjoying living in the 13th century, and adding nothing of value to knowledge or understanding.
- Hans Georg Lundahl
- [Only potentially so.]
« Meaningless. »
Again, to your barbarous and clumsy modernity of thought. No, quite meaningful. God knows the whole potentiality into infinity of the Fibonacci sequence – but He also knows very well which of the ratios He did and which He didn’t give actuality by creating things with sizes or numbers or timelengths etc with the ratio.
[Let us take an example of limes and sequence, shall we.]
“What is a lime?”
Limes is the Latin nominative singular for a stem which goes “limit-“ in all other cases - and in English it seems the loan word identical to the [Latin] stem is as such used. In German or Swedish you can’t quite do that. “Grenze” or “gräns” is limit or frontier. It is not a Mathematical term. In Maths you either say “Grenzwehrt” or “gränsvärde” or you use the word as in Latin, not as in English. Limes. Too bad for English Mathematicians if they can no longer use the Latin term.
Lime is a fruit I was not alluding to either in singular or in plural, even if the English plural of lime is homographic with the Latin singular limes.
[However long the sequence may go on in the operation of the mathematician, it will certainly converge with but never reach the limes φ.
Replacing the finite values of sequence with infinity-of-that-sequence is really stepping out of the sequence.]
“Who says that it will "reach" φ? What is "infinity-of-that-sequence"? What is "stepping out of the sequence"? It's like you are inventing things that no mathematician does to justify your stupid position.”
Well, if you admit it is not reaching φ, you have admitted the sequence is never drawn out to infinity in actuality. Which is you have admitted my position as correct.
In that case my attack is not on what modern Mathematicians are doing, just against the shortcuts of their terminology which camouflages it to non-Mathematicians.
But when Kant could believe Universe was a round disc infinite in two dimensions and limited only at straight angles to it, well, you can see how the wording you used may have done mischief in other fields. Which are my main concern.
[My justified reason is observation of mathematicians at every turn of their real practise.]
“Well then, you seem to have little experience of working with mathematicians.”
This video justified it again. I hope I have missed no video of numberphile dealing with “infinity” and my empiric evidence of watching those seems to point to St Thomas having very accurately observed geometricians in his day and mathematicians have not changed practice since, just gone sloppier in terminology.
[In the sequence of fractions that man did not go on into infinity, he stopped at a very early level of execution of the series.]
“What do you mean by "execution of the series"? You are talking meaningless, non-rigorous babble.”
If by rigorous you mean translatable into formulas using symbols without words, you are right about my talk, except in calling it babble (unless you are the businessman of Athens who called Socrates a babbler when he condemned usury – in which case you are right from your point of view). But if so, I hate to have been so late in bringing this to you, but you have a wrong sense of what logical rigour means.
The series could have been executed, taken out, performed, brought to actuality, explicitated etc. another step. And another step. And so on, POTENTIALLY into infinity but NEVER ACTUALLY reaching it.
The problem with your trying to express everything in symbols is or includes that there are no symbols for the distinction between potential and actual. There are however Thomistic concepts of it.
- jamma246
- You just prove that it's impossible to reason with a religious person. If people kept holding to your views, we would still be in the dark ages.
I can't be bothered to talk about this anymore, or read your huge ranting posts.
Keep reading!
- Hans-Georg Lundahl
- My so called rants are answers in detail to what you wrote.
Glad my last one stopped at 618 words.
And glad you show your colour as not interested in maths as such or mathematical truth, but in promoting secularism.
If it is impossible for you to reason with a "religious person", it may be because you are yourself not very worth reasoning with.
I started enjoying the thing when you gave examples, and was planning to satisfy you on the "according to what rule" question too if rechallenged.
There used to be a time when infidels were worth either running a sword through OR reasoning philosophy with. You are not promoting a preference for the second alternative.
- jamma246
- "And glad you show your colour as not interested in maths as such or mathematical truth, but in promoting secularism."
I am extremely interested in maths and mathematical truth. That is why I reject your non-rigorous mumbo-jumbo. And, by the way, secularism is about allowing all relevant positions to be heard. That doesn't mean that one has to accept irrelevant antiquated opinions that are not rooted in logical sense.
"I started enjoying the thing when you gave examples, and was planning to satisfy you on the "according to what rule" question too if rechallenged."
That's the point. The constructions work within the confines of the theory. For example, there is a world of maths where the Axiom of Choice holds. There is one where it doesn't. But neither is "real" or "non-real". At that point, where you are segregating "real" maths from "non-real" maths, you jump from mathematics to pseudo-science, and that is precisely what you are doing. And mankind has made great progress since it dropped such naive techniques.
- Hans-Georg Lundahl
- "I am extremely interested in maths and mathematical truth."
Just the other day you said mathematics is at heart just a bag of tricks.
To me that is a very true obervation of ALGEBRA, but not of ARITHMETIC, nor of GEOMETRY.
"That is why I reject your non-rigorous mumbo-jumbo."
I am not sure exactly what kind of rigour you demand, I hope it's not rigor mortis. I suggested you meant by rigour sth which can be shown using symbols of mathematical convention ideally without using words, except the initial ones explaining each symbol.
If that is so, you have a very different attitude - a really superstitious, fetischistic mumbo-jumboish - to what scientific stringence or rigour ought to mean. From what St Thomas had (in his time to make an addition you had as yet no +, you used a. = adde), and from what I have.
"And, by the way, secularism is about allowing all relevant positions to be heard. That doesn't mean that one has to accept irrelevant antiquated opinions"
In other words, secularism as you define it is about allowing all positions about a thing to be heard except the one which is most relevant, because the true one.
Old does not equal antiquated. 2+2=4 is as old an observation as Adam and Eve, way before any mathematicians did things. Does this mean it will one day be antiquated? Of course not!
You show off the relevance of the Thomistic position, not by admitting it verbally, but by harping on its irrelevance (to your set of mathematicians that is no doubt a subjective truth) rather than using arguments why it should be considered faulty.
"that are not rooted in logical sense."
The Thomistic positions as a collection in general and those I gave on mathematics are on the contrary very well rooted in the logical sense.
You see, Arithmetic does deal with a central Thomistic concept, namely one, taken as an undivided whole, as opposed to its multiples.
And Geometry deals with another side of same concept, namely one, taken as the whole and divisible.
It is on the contrary algebra which is not rooted in the logical sense.
Saying that "-b" * "-b" somehow "= + b^{2}" is very right per accidens in certain contexts, but very wrong if taken as a general rule.
(a - b)^{2} per se = a^{2} - ab - b(a - b) - and sorry if I gave the wrong amount of subtraction for the second one last day, I was a bit flustered - but only per accidens does it = a^{2} - 2ab + b^{2}.
You see, - is simply a new sign for "s." as in "subtrahe". And by subtracting from a subtrahend you end up subtracting less from the minuend, that does not mean you actually add to it. Some have, with very shallow grasp of logic or very subtle disregard for this reality considered the algebraic rule of "-b * -b = +b^{2}" as being a truth in its own right rather than an aspect of the larger truth which per se is expressed "(a - b)^{2} = a^{2} - ab - b(a - b)". And a minor aspect at that, since a - b = c, a subtraction must leave a difference. A real subtraction can never have zero or "less than zero" as such a difference.
Up to about 1600 there was no unity among mathematicians (whatever unity there was afterwards, up to now) on treating zero as anything but a place holder in positional Arabic numerals. Between 1500 and 1600 (very roughly) there was a debate. It ended, alas, on the wrong note.
"The constructions work within the confines of the theory."
Saying even "(a - b)^{2} = a^{2} - 2ab + b^{2}" is not so much theory as convention.
"For example, there is a world of maths where the Axiom of Choice holds. There is one where it doesn't."
I do not know what the "Axiom of Choice" means even. It may be a very basic truth or a very obvious non-truth, but one of the worlds of math you refer to is not real.
"But neither is "real" or "non-real". At that point, where you are segregating "real" maths from "non-real" maths, you jump from mathematics to pseudo-science, and that is precisely what you are doing."
I have no shame in having my very defensible positions labelled pseudo-science by a modern academician whose own activity is hardly strictly logical.
As to your last words, "progress" is vastly overrated.
I came into Creationism (maximum six literal days and probably that maximum as diverging from the option given by St Augustine in De Genesi ad litteram books V and VI), into Geocentrism, into Angelic movers of the celestial bodies and into Flood Geology before I came to start debating with mathematicians or their adherents among my opponents about more than ten years ago.
So, if you think you can intimidate me by indignation outbursts about my doing "pseudo-science", you are barking up the wrong tree.
Take the debate on grounds of shared mathematical facts and logic, or leave it. But lecturing on "progress" and on "pseudo-science" is simply just annoying.
- jamma246
- "Just the other day you said mathematics is at heart just a bag of tricks.
To me that is a very true obervation of ALGEBRA, but not of ARITHMETIC, nor of GEOMETRY."
This seems to be my point. I don't actually regard mathematics as a bag of tricks, that was a bad explanation, but what I meant is that it is all abstract, there isn't abstract vs. non-abstract mathematics, that is a useless and ill-defined philosophy. Mathematics is all about logical deduction. Your position seems to be that some mathematics is "real" (e.g., it seems algebra) but that some is not (e.g., arithmetic and geometry).
I simply don't understand how or why you make this distinction. Why, for example, is Euclidean geometry something which you see as "real" mathematics, that "exists" and not algebra? It is very likely that the world that we live in isn't even based upon Euclidean geometry, so your position seems to be meaningless.
Anyway, you have proven yet again how religious thought can lead one astray. I cannot be bothered to comment any longer.
- Hans-Georg Lundahl
- "I don't actually regard mathematics as a bag of tricks, that was a bad explanation"
If anything a bad statement of your position, not sure your positions is fixed enough for you to know it.
"but what I meant is that it is all abstract,"
I agree, whereever it becomes concrete like a geometric figure in a construction or a geometric occurrence to be studied, it is no longer pure but applied mathematics.
A 3:4:5 triangle is abstract, make one on a paper and it may be for instance 9.42 cm : 12.57 cm : 15.71 cm, which is its concrete size adding nothing to the abstract concept of a 3:4:5 triangle.
So, yes, mathematics as such is always abstract.
"there isn't abstract vs. non-abstract mathematics, that is a useless and ill-defined philosophy."
I have never said there is abstract vs. non-abstract mathematics, unless now if by non-abstract you mean the concrete sample studied.
I have said some abstractions are from real concrete objects having something in common, and some of these objects are of mathematical nature, like multitudes and magnitudes, and so we get real abstractions like arithmetic and geometry.
Other abstractions are rather from the universally observed "behaviour" of the concrete objects and, though licit, rather constitutes fiction than science.
Never said you can't have a numberline (with zero and "minus numbers"), but just that unlike the series 1, 2, 3 it does not accurately depict a series of multitudes.
So, not "some mathematics is abstract, other is concrete" (excepting of course distinction of discipline from its samples), but that some abstractions are based on reality and others, more or less usefully, are fooling around with it.
I am not into denying anyone the pleasure of fooling around with reality, love reading Tolkien myself, but you seem to be taking:
- either x=5-7 for as real (though as abstract) as x=7-5
- or x=7-5 for as abstract (and hence as unreal?) as x=5-7.
THAT is bad philosophy. The counterpart would be to take either LotR for as real as Anglo-Saxon Chronicle or Anglo-Saxon Chronicle for as fictional as LotR.
"Your position seems to be that some mathematics is "real" (e.g., it seems algebra) but that some is not (e.g., arithmetic and geometry)."
On the contrary. Euclidean geometry is real (except the fiction - useful as it is - that lines are infinite), so is Pythagorean arithmetic, so is their tête à tête in Boethius and in his inheritor St Thomas.
"Why, for example, is Euclidean geometry something which you see as "real" mathematics, that "exists" and not algebra?"
I have given examples of how algebra uses rules and in some cases end up with concepts having no concrete counterparts in reality - i e constituting malformed abstractions. (sqrt of minus 1 is a favourite example).
You defend your position they make as much sense as Pythagorean theorem if you like, but you changed the subject to make a tirade, a frontal attack ad hominem about my attitude to mathematics.
One thing you may have said very truly! You "simply don't understand how or why you make this distinction," that may possibly be true, you may possibly have bent your reason to disregard obvious distinctions that much.
Not my fault, not my problem.
[Unless someone tries to make it so? Then that kind of acting would be my problem.]
"It is very likely that the world that we live in isn't even based upon Euclidean geometry, so your position seems to be meaningless."
That likelihood is a very moot point.
It is for instance possible that modern maths is deluding modern physics and modern physics is deluding modern maths.
Even so, Euclidean geometry is a phenomenon which very obviously does occur in the world as we see it.
It is just that Euclid analyses examples which are so simple they do not really occur, because these simple examples illustrate what does occur.
"Anyway, you have proven yet again how religious thought can lead one astray."
You are proving again you are pushing secularism rather than logically defending your take on maths.
There was NOTHING in my reasons for this position which you could pinpoint as religious. Saying my preference for maths understanding of Boethius and St Thomas is Catholic biassed is like saying someone else's preference for Einstein's physics is Jewish biassed.
In fact Boethius and St Thomas share their understanding with two pagans (Euclid and Pythagoras) each of whom also had a different religion not just from the Catholics but even from each other.
"I cannot be bothered to comment any longer."
Is that a promise?
- either x=5-7 for as real (though as abstract) as x=7-5
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