Are finite numbers an assumption in mathematics?
I define infinitism as the belief that only infinite numbers exist, within which individual units don't mean anything apart from the whole. Is the idea then that 1 plus 1 equals two a philosophical assumption of mathematics or is it the only way to understand mathematics? Any finite number can be seen as having infinite units within it and so 1 plus 1 equals two would mean one and one infinite numbers equals two infinite numbers, which this is not correct except as a tautology of some sort. So does mathematics have dogmatic assumptions or is it really a rigorous discipline? Thanks
Comments (79)
Well addition subtraction and the rest mean something for mathematicians, but if everything that is real is just infinite sets a lot of mathematics would become frivolous or unnecessary. My current position, I admit, does deny that numbers are real, but I think I have a point in that there could be a certain mathematics only of the infinite. I add and subtract for practical purposes but there doesn't feel like anything Platonic or ontological about it for me
Do you think mathematics of the infinite can be done without finite numbers?
I think of units of an infinite set as completely relational to the whole.
You're more educated in this than I
Whoa! That is quite an assumption. :scream:
I don't mess with infinities. Others here do.
I read a lot of German idealism and I strongly feel they were arguing against the possibility of finite existence. Mysticism usually subsumes everything into infinities
Mathematics is rigorous by effectivized formal languages, recursive axiom sets, and recursive inference rules, and explicit statements of algorithms for checking for well-formedness, axiomhood, and proof.
Mathematics has both finite and infinite sets.
Your notion of a mathematics with only infinite sets is purely fanciful without at least an outline of the primitives and axiomatic notions.
Well this is a philosophy forum, not a math forum, so we need to try to stick to the basics of the philosophy of mathematics. Stick with me and read the following:
"It is thus not surprising that Hegel's book began with a devastating, even if very ironical, critique of Jacobi's position against Kantianism (and all the forms of post-Kantianism), namely that we are in possession of a kind of 'sense-certainty' about individual objects in the world that could not be undermined by anything else and which showed that there was an element of 'certainty' about our experience of the world that philosophy was powerless to undermine. Hegel called this a thesis about 'consciousness.' If we begin with our consciousness of singular objects and present to our senses (an awareness of 'things' that is supposedly prior to fully fledged judgments), and hold that what makes those awarenesses true are in fact the singular objects themselves, then we take objects to be the 'truth-makers' of our judgments about them; however in taking these objects to be the truth-makers of our awareness of them, we find that our grasp on them simply dissolves and the impetus for such a dissolution lies in the way we are taking them to play a role in consciousness. The result, Hegel argued, is that in the process of working out these tensions, we discover that it could not be the singular objects of sense-certainty that had been playing the normative role in making those judgments of sense-certainty true, but the objects of a more developed, more mediated perceptual experience had to have been playing that role all along... The dialectic inherent in Jacobi's sense-certainty thus turns on our being required to see the truth-making of even simpler judgments about the existence of singular things of experience as consisting of more complex unities of individual things-possessing-general-properties of which we are perceptually and not directly aware. That is, we can legitimate judgments about sigfular objects only be referring them to our awareness of them as sigfular objects possessing general properties, which in turn requires us to legitimate them in terms of our take on the world in which they appear as perceptual objects... We must acknowledge, as Kant put it, that it must be possible for an 'I think; to accompany all our consciousness of things." Terry Pinkard
So I am not trashing mathematics but, instead, probing it's assumptions. The view of many philosophers is that infinities alone exist and I'm wondering what happens to the rest of mathematics when this is accepted by someone. Infinities have cardinality and density, the former perhaps being bound to the latter and perhaps geometry as well. Hegel in particular thought mathematics eventually turns into theories of the infinite and pure logic and I thought it would be interesting to see other peoples' take on these questions
Sure, in a purely philosophical context, you can come up with all kinds of stuff.
But then mathematics is not one hundred percent truth. Even nowadays infinites things haven't all been worked out fully. Imagine a ruler going to infinity out into the horizon and into space forever. Is the ruler longer as a whole or equal to the odd sections? This illustrates that the relationship between cardinality and density can be tricky but then it sublates into a higher view that nothing is ultimately "longer" in a non-dualist ontology.
Sometimes novel ideas are helpful. I tried to start a discussion once about what it would mean if all math was wrong and the *opposite* of every equation and theorem was true. People didn't like that idea but I tried this thread out anyway. I recommend the essay "Holism and Idealism in Hegel" by Robert Brandom
What do refer to when you say 'mathematics'?
The theory of numbers. 1, then double one, ect. But a non-dualist or infinitist mathematics would no longer deal with number and I admit I don't have many details worked out, but I think it's an interesting idea in that it would think solely in terms of concentric infinities instead of finite units
They're not a doubling sequence. They're a successorship sequence.
And what theory of the natural numbers? There are many.
It's just about numbers in general vs sets which have infinite density
That sounds exciting! Sorry I missed it. :lol:
So anything anyone says about numbers in general is mathematics? And if your philosophizing about that mathematics is purely philosophical then that mathematics is not all true?
And what do you mean by 'infinite density'? Is that a mathematical notion of yours or purely philosophical?
Sometimes there is disenchantment with nature in math and science. But nature is truly enchanted. There is no such thing as a disconnected observer and the opposition within the dualism of "how things are" and "how I perceive them" can be overtaken by a holism in how we take things,
From what I understand the terms "ordinal" and "density" are very much related. Wikipedia has articles on these terms but I don't know if you want to read into the math on this or not. HOWEVER, the philosophy of mathematics is a real field of study. I have books on it, and that is what this thread was about
Logically proving mathematics from the ground up was the incompleted project known as Logicism
And I know that the philosophy of mathematics is a field of study.
Logicism is often thought to have failed because it was not found how to derive mathematics without non-logical axioms.
That does not refute what I said about the rigor of mathematics.
Imagine the ruler going into space infinitely again. The density of the whole is greater than the odd parts of the ruler but the cardinality is the same. Take a marble and the Earth now. They have equal number of parts in that they have the same uncountable infinity of subdivisions. But which is larger? My point is that math should rise our thoughts to higher stages by these thoughts instead of getting stuck in an infinite process of proof and review, especially considering Gödel's theorems. Math is not done apart from Nature. What do we mean when one thing is opposed to another?
I find no definite sense from your use of all that undefined terminology and assumptions. And I surmise that continuing to ask you will lead to only more.
But I am intrigued what you have in mind regarding Godel's theorems.
Ok. Mathematics is usually practiced as a Platonic type of religious practice. What did Plato espouse? Well he though there was a partition or doubling in reality with the shadow of the world on one side and intellectual reality on the other. These are one and yet two (very unmathematical in that sense) where one is the other of the other. I just see so much separation from the world with the union of math and science that I had to find Naturphilosophie to make sense of it. It's been widely reported that Neil de Grasse Tyson has caused a lot of depression in people (lol). One alternative is to say teleology is given by the divine to the world but another perspective is that the world is teleology and we are Nature. Anyway thanks for you insightful posts
No, it's not.
If they don't then they approach it from a dead universe perspective and will move in infinite circles (circles which, by the way, are better understood as infinites within Nature)
A lot of those into science believe matter is dead unless organic. But nothing is dead and I'm not talking about pansychism here. I'm taking about a world that has beauty and which responds to our ideas. Scientist often escape into a Platonic world and more than in other fields have breakdowns, as Gödel and Cantor unfortunately had. I started this thread with a little talk about numbers and have now moved on to my larger program about sickness of mind in the West
Darn, I was hoping to hear about a work in mathematics that is from a dead universe perspective and moves in infinite circles.
And now I'm also hoping to hear of a study that shows that scientists have an especially high incidence of mental illness.
Then I was wondering what you had in mind when you mentioned Godel's theorems, but the only thing so far you've mentioned is his and Cantor's breakdowns. If you are suggesting that Godel and Cantor suffered mental illness due to their work in mathematics, then that would require evidence. Also, I hope you are not suggesting that Godel's and Cantor's personal problems discredit the mathematics.
I had a colleague, a fellow math prof, who was a devout Catholic. I once asked him what he thought it was that we do as mathematicians. He said, "We are like priests who practice their faith, it's what we do."
Mathematicians do sometimes say cheeky things like that. But mathematics doesn't have religious practices, rites, rituals, creeds, obligations of obedience, eschatology, myths, scripture, moral strictures, determination of truth by fiat and divine revelation, etc.
Mathematics is the opposite of religion.
Au contraire, he was right!
Gödel already proved that mathematics is either wrong or that there are infinite things that can't be proven. The problem is that they can never find where the line is between what is provable and what is not, so mathematicians will forever be searching for the finite sequence of proofs that are certainly knowable. It's enticing to look for total certainty in your field of study but it can cause problems
We can list the essential attributes of religion, and see which of those are attributes of mathematics. Attenuated, glib, "kinda sorta" analogies like "axioms are a fiats" or "infinity is mystic" don't count.
You seem overly rationalistic
That is one of the worst encapsulations of Godel I've ever read.
Quoting Gregory
Another botched attempt.
No no no. It's what you feel, not what you know. You are thinking like a logician. :gasp:
No it's correct. I stated it boldly to bring out its true nature
Just to be clear, that ad hominem doesn't refute anything I've said. Second, I have all kinds of facets, not just reason, but in discussing mathematics, I prefer to be lucid.
You responded to the wrong person and wrote like a Logicist
If I understand you correctly, sure, one can feel that one is being religious. I am not disputing that he is accurately describing the way he feels or even views what he does.
You are expecting too much of us. We are mere mortals, and few of us commit ourselves to this challenging task.
I feel like you ask for proof for the obvious
No, I didn't.
Quoting Gregory
That's idiotic. Logicism is the view that mathematics can be derived from logical axioms alone. I've never even flirted with logicism.
What ad hominem?
Where did Godel say it?
Ah, argument by "it's obvious".
Your claim that I seem overly rationalistic, obviously.
In his 1931 paper he said he set out to prove that mathematics was inconsistent (aka wrong) or incomplete. Where can the line be drawn with which to distinguish the provable from the unprovable? They still have to work out the foundations of mathematics. It sounds like you make logic or math a religion
I said "seem" which means "appears to others' perception"
And that doesn't deserve the mangled version you posted earlier.
Quoting Gregory
It's not clear what "line to be drawn" means there. And every formula is provable in some system or another.
Oh please, a minor pedantic shift in the way one casts an ad hominem doesn't erase it's affect as an hominem.
Of course I don't. I listed a good number of the major features of religion. I don't relate to mathematics with any of those features.
I merely repeated my previous post on Gödel. I just reread both of them. Anyhow are you saying they can find a limited number of theorems that they know for sure is provable from the ground up?
"Quoting Gregory
and
Quoting Gregory
are critically different.
If you find the word "rationalistic" insulting, it be interested to know your world view.
I think you are being too literal. I am working on a conjecture right now, as I have on others on and off for fifty years, with a sort of pleasing fervor that could be described more as a religious mood than a purely problem-solving commitment. Its true than mathematicians can be very competitive, even mean spirited at times, and I have been in an unpleasant situation that reeked of it, but exploring math - rather than learning it - is a delight and has its rewards.
Godel discusses formal theories. What does "provable from the ground up" mean in regards to Godel's incompleteness theorem?
Those two sentences together comprehensively state Gödel system without using equations
It was not 'rationalistic'; it was 'over rationalistic'. And it's not that I find it so insulting, but that it is ad hominem.
Well let's put the rationalist thing aside. How do you know numbers have truth value? Do you base mathematics on philosophy or on itself?
Too literal about the word 'religious'? It is used as cudgel. So it is good, as a starting point at least, to point out that it is not literally true.
The second one is reasonable. The first one is a mess.
Whoever said numbers have truth value(s)?
1 plus 1 equals two only if 1 and 2 exist and can exist. So things boil down to our world view at the end of the day
I asked whoever said that numbers have truth value(s)?
Or is it your own claim that numbers have truth value(s)?
All mathematicians says that numbers exist in some sense, which is the same as saying they have truth value. The struggle over the foundations of math is a search for ultimate certainty, much like Descartes's journey in 1641-42. Searching for logical certainty can be a mask for too much doubt
No, what have truth values are statements.
What have a truth values are the statements "There exists a natural number that is the successor of 0" and "There exists a natural number that is the successor of 1".
The numbers themselves don't have truth values.
This is not a mere pedantic distinction. It is a distinction needed so that the discussion about this subject is coherent.
Quoting Gregory
The theorems are certain or uncertain exactly to the extent that the axioms are certain or uncertain.
It is freely granted that the common axioms are non-logical. That is why it is really stupid for you to say that I'm a logicist (or however exactly you said it).
However, if any non-logical mathematical judgements are certain, then they are those of finitistic combinatorial mathematics. We may be skeptical of finitistic combinatorial mathematics, but then we might as well be skeptical of everything mathematical.
Words and thoughts about numbers are just like any sentence in language and they depend on how much the person values them. Consider that divination is the same as reading philosophy. They are a reading of something through a perceptual experience of our being of-the-world. The only issue is how much knowledge is gained and what is useful for life
For a consistent formal (I always mean 'consistent formal' in this context) system S, there is an infinite enumeration of the proofs. So it is linear, not circular.
In a idealized context without a finite upper limit of time to prove (such as with Turing machines), if S is incomplete, then for a sentence P, at any point in the enumeration, there are these possibilities (proof relative to S):
(1) P been proven.
(2) ~P has been proven.
(3) P is provable, and we will eventually find a proof.
(4) ~P is provable and we will eventually find a proof.
(5) P is not provable and ~P is not provable, and we will never find a proof of P and we will never find a proof of ~P ("eternally floundering to know" whether P is provable and "eternally floundering to know" whether ~P is provable).
(6) In a metatheory for S, we prove "P is provable or ~P is provable".
(7) In a metatheory for S, we prove that P is provable though we don't know a proof itself.
(8) In a metatheory for S, we prove that ~P is provable though we don't know a proof itself.
(9) In a metatheory for S, we prove that P is not provable. (e.g., Cohen proof that AC is not provable from ZF and that CH is not provable from ZFC)
(10) In a metatheory for S, we prove that ~P is not provable. (e.g., Godel proof that ~AC is not provable from ZF and that ~CH is not provable from ZFC)
(11) In a metatheory for S, we prove that P is not provable and that ~P is not provable. (e.g., the conjuction of Godel and Cohen)
For mortal beings, or assuming that in some finite time there won't' be any conscious beings, "eventually" does not hold.
But mathematics itself is very clear about the limitations mentioned in (1)-(11) and makes that lack of conclusiveness itself a subject of rigorous study. This is yet another respect in which mathematics is diametrically different from religion.
I think there are infinite things we can prove with math and infinite things we cant. But math starts with assumptions and that was my point. There are philosophical ways of understanding reality that excludes mathematics and all the assumptions connected to it that sprang from the school of Pythagoras. I'm not saying do away with mathematics. I was offering a way for people see reality that is based more on classical Romantic thought and dialectic than anything traditionally mathematical
For any given consistent system S, there are infinitely many theorems and infinitely many non-theorems.
Yes. I was just offering a philosophical look at numbers
If you read this carefully, you will find a new type of thinking that is different from the tradition of Kant and his predecessors. It's all about logic, infinities, and philosophy