Looks like an error in your truth table for implication. F => F is T.

But I haven't been following the discussion and you may be into something else.

Comment: The identity $\sqrt{x} = 1+\frac{x-1}{1+\sqrt{x}}$ generates a periodic continued fraction

$\sqrt{x}=1+\cfrac{x-1}{2 + \cfrac{x-1}{2 + \cfrac{x-1}{2+{\ddots}}}}$ from which $\sqrt{2}$ can be calculated by iteration. This might be the algorithm used to obtain square roots on simple calculators. Or it may have been some time ago, replaced by better algorithms. CS people out there?

This expansion may be due to Omar Khayyam, the poet, rug maker, and mathematician from around 1100AD. :cool:

Human beings may have gotten over this, but they did not resolve the problem — Metaphysician Undercover

. . . "Two dimensional objects have a fundamental problem which demonstrates that space cannot actually be represented in this way . . . We see a very similar problem in the relation between zero dimensional figures (points) . . . Then we can see that it is only when we apply numbers to our dimensional concepts of space, that these problems occur. . . None of these numbers systems has resolved the problem because the problem lies within the way that we model space. . . . The problem though is that introducing real numbers does not actually solve the problem, it just offers a way of dealing with the problem."


Forgive me, but what is the problem, again? :worry:

But perhaps someone reading this took a course in groups, rings, and fields but forgot this beautiful construction, which we can sum up in one equation: — fishfry

My very first course in abstract algebra (taken in my first semester in grad school) did something like this. Not being conversant with the various concepts, even groups, made it very challenging and also meaningless. Afterwards I took a course in group theory which was illuminating. Thereafter I avoided abstract algebra. :brow:

A mistake, looking at current complex variable theory!

Good luck with this project. :roll:

I always thought this was an abstraction of basic 17th century calculus, where higher powers of infinitesimals can be ignored. — fishfry

True enough. In computations involving non-infinitesimal calculus higher order terms can be ignored depending on the settings.

I know that in constructive math, all functions are computably continuous or something like that. Makes some of the problems go away. — fishfry

Hmmm. I keep learning things here. Thanks. :chin:

https://people.eecs.berkeley.edu/~fateman/papers/limit.pdf

Some years ago the New Math was in vogue. As a mere instructor at the time I was given a text on College Algebra having a lengthy first chapter devoted to an axiomatic approach to the subject. It was not a good experience for instructor or student. :worry:

You can learn something on this forum. A number of years ago I bought Robinson's book on NSA and got excited about it for a while, even contemplated teaching calculus that way, but put in on a shelf instead and moved on to other ideas. But I had not heard of SIA until joining this forum. I can't say I am very impressed with it, however, not really wanting all my functions to be continuous and all! And the defining characteristic involving squared infinitesimals seems just another strange notion one can avoid. :wink:

I've considered it. It uses a complex variable for time. That is unlike any time I'm familiar with. So I do not think it reflects the universe we live in. — Devans99

I think physicists resort to complex (or imaginary) variables when it is convenient to do so and by doing so can predict phenomena. There's no magic or mysticism or metaphysics usually, just a path forward that produces results. The Feynman path integral uses a complex integrand, for example. :chin:

We now know that the moon is demonstrably not there when nobody looks. — Boojums All the Way Through - N. David Mermin

The movie, The Time Machine, taken from Wells' novel, shows the moon that is not there breaking apart in a catastrophic sequence, which, apparently, is not there as well. :scream:

Good explanation. I seem to recall from long ago a study by Piaget on the ability of young people to understand calculus. I may be mistaken but it seems that in general the age of fifteen was a benchmark, with those below that age experiencing a lot more difficulty with the subject. Of course there are spectacular exceptions. As for the intricacies of the real number system, I wonder. :chin:

I don't really believe in multiple universes, but if they do exist, then which of the following is more likely:
1. They are all made of completely different stuff and evolve in completely different ways
2. They are all made of similar stuff and evolve in similar ways
I think the 2nd is much more likely, leading to the conclusion that most or all such universes support life; a conclusion that fatally undermines the so called strong anthropic principle. — Devans99

Since MUs is such a mind-blowing concept I don't think "more likely" has any bearing. But have a good new year! :cool:

Blatant contradiction is not the real problem though, rather ambiguity and vagueness, such as the difference between "continuum" and "continuity", the definitions of "object" and "infinite" are the real problem. — Metaphysician Undercover

It's mostly a metaphysical problem. Most mathematicians and physicists do quite well without contemplating such issues. But that's not to say that "infinity" and "objects" are not concerns, as a physics person might tell you in reference to renormalization procedures and quantum entities, for instance.

Each of those countless universes is made of the same stuff and evolve in the same way, so they all support life — Devans99

How can you possibly know this? Have you traversed the spectrum?

But, interesting thread. :chin:

And then there is determinism.

Wiki: "Confusion of causality and determinism is particularly acute in quantum mechanics, this theory being acausal in the sense that it is unable in many cases to identify the causes of actually observed effects or to predict the effects of identical causes, but arguably deterministic in some interpretations"

Also

Wiki:"The many-worlds interpretation accepts the linear causal sets of sequential events with adequate consistency yet also suggests constant forking of causal chains creating "multiple universes" to account for multiple outcomes from single events."

This is where I become interested in metaphysics, for I favor this concept. :cool:

And maybe he's reformed his ways and is now writing articles of actual substance. — fishfry

:lol: Not likely. I thought the thread needed some comic relief.

well we can argue what x consists of all year but x is still x — Mac

And this is true of entities that defy the winds of time, like 2=2. But introducing a time gap when speaking of physical objects is different. X(t1)=X(t2) ? Is one PM today the same as one PM tomorrow? Is anything physical the same from day to day? One could argue that my car yesterday is the same as it is today, in a rough sense, but of course it isn't.

Must we pick at the bones of this troubled man?

Wiki: Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I.[33] The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life.

Profound set theory can be harmful to one's health, Devons99!

Time is not a thing, so dividing it is rather arbitrary — Gregory

Is a line segment a "thing?" As opposed to a wooden rod, say? Does a line segment actually exist as a physical thing?

we have gotten new insights on mathematics in history and our understanding of math has greatly changed from what it was during Ancient times and what it is now. Hence what is preposterous is then to think that a) no new insights will be made in mathematics in the future and b) these new insights won't change our understanding from the one we currently have. — ssu

Not only do the foundations shift, but mathematics rolls along like a giant intellectual snowball, gathering layer after layer of new concepts and theory, a plethora of results that can be bewildering even to an expert in a specific area. I was in a classical area, complex analysis, for years, and still dabble in elementary research, but these days I can hardly understand the titles of papers in that subject.

Perhaps Peter Lynds' essay on the impossibility of "points" in time has appeared in this forum. If not, some might be interested in his discussion of time's continuity.

https://arxiv.org/ftp/physics/papers/0310/0310055.pdf

This paper inspired considerable controversy. You be the judge. :chin:

Therefore you can make a universe out of a pebble — Gregory

Assuming God has laid forth unto us the Axiom of Choice.

The conclusion is that the limits of calculus go out the window!
I've merely had the courage to take what Metaphysician Undercover is saying to it's logical conclusion — Gregory

This is quite entertaining. :smile:

At the same time, the university seems to be hellbent on getting first-year freshmen to "study" gender studies, . . . — alcontali

Political correctness is something I do not miss since retirement. :brow:

The main point is that this is how you define cardinal numbers these days. They're no longer equivalence classes of sets that themselves aren't sets. That was a problem so it got fixed at the expense of needing to do some technical work. — fishfry

Interesting. Thanks. Nice exposition. I only took one course in set theory almost sixty years ago and it was Halmos' Naive book. When he started discussing towers and chains it seemed medieval and it was hard to stay motivated.

I recall seeing that problem about [0,1] and [0,1) on a PhD exam some years ago. I don't think I would have gotten it in the time allowed! :worry:

Your book is very impressive. And your posts are excellent. :cool:

That is why majors in almost any subject will graduate with close to zero understanding of that subject. The only ones who understand the subject are people who have been confronted with solving practical problems in that subject. Everybody else invariably sounds like an idiot. — alcontali

This seems a bit harsh and I do not agree. However, I will admit that working in an area may clarify and solidify the knowledge gained as an undergraduate. In the academic world the problems don't necessarily have to be practical to have this effect.

I understand that this is generally necessary for university students as they simply don’t have time to read through anything themselves. — I like sushi

Yes. However, he implied that anyone beginning a study of a particular philosopher should read not only those works, but other's critiques as well.

where a field is by definition a mathematical object which has a value everywhere, so "field" is almost a synonym for "plenum" — Pfhorrest

Nice exposition. A few additional comments: There are various kinds of fields. A mathematical field is more an algebraic structure with operations, like a group or ring. Complex or Euclidean vector fields have associated with each point therein a direction and a magnitude. A force field is an example. A time-dependent force field has the property that each point has a variable force vector that varies with time. Shifting magnetic fields are an example. Most of space, I imagine, has some sorts of fields running throughout, without meaning a material substance. On the other hand, plenum might refer to the aether (if you think it might exist).

Monads now exist as proper mathematical entities in non-standard analysis.

(I once searched my math pedigree and found that Friedrich Leibniz (father of GWL) was my earliest ancestor)

I think the notion of an orange being divided infinitely is ap-peeling. :smirk:

I'm so happy that Cauchy, Weierstrass, and others settled this issue for mathematical analysis long ago. It made my career so much easier. :cool:

I also think it is why the ability to grasp the meaning of the equals sign, that X=X, is essential to the formation of intelligible ideas and language. — Wayfarer

Well, you have to start somewhere, I suppose.

What do you think of X=X+1 ? :gasp:

By this I mean that you should avoid this the first time around and come to your own conclusions about the text written by the philosopher before being spoon fed someone else’s interpretation. All philosophers are basically working from others anyway so why bother to distance yourself fro the text by seeing it through the lens of another? I understand that this is generally necessary for university student — I like sushi

My knowledge of philosophy is limited, but I recall my prof giving the opposite advice after I tried reading several of the well-known philosophers like Kant in their own words. Some of those icons wrote poorly.

This is distantly connected with wave/particle duality in physics. But when an electron, for example, is detected, it is detected as a particle. Whereas its wave form is a probability wave detected by slit experiments. Does everything "exist" as a wave form? Yes, I suppose, but at larger sizes that's not a useful approach. (Calling a real, live physicist if there is one around this forum - correct me!) :chin:

Zenos paradox shows the infinity within the finitude of objects. The fact we can break a candy bar in two shows this applies to our world. That is enough for banach tarski. You can take infinity out of infinity. The extra cantor stuff well extra — Gregory

Once again, I can't argue with this. :brow:

There are only two classes of people who need to carefully make this distinction: mathematicians, who are trained on this topic in their undergrad years; and philosophers, — fishfry

Mathematicians? Not necessarily. "Flaws" . . . not necessarily. Incidentally, your compact form of Leibnitz expansion has a simple error. And 1/3 =.333... = limit of a geometric series, well defined. You may be talking about mathematicians who labour in foundations. Making such fine distinctions is unnecessary in most math careers, IMHO.

Zeno is a shortcut to Banach Tarski — Gregory

So how does Zeno produce a non-measurable set? :chin:

The axiom of choice results from the infinite divisibility of objects — Gregory

So the AC is a logical consequence of infinite divisibility and thus is not an axiom? :gasp:

I don't see why Zeno's paradox is not a paradox but Banach-Tarski is. The latter flows directly from the former, and there is no BT without Zeno — Gregory

I don't know about that, never having gone through the proof of B-T. However . . .

Wiki: Unlike most theorems in geometry, the proof of this result depends in a critical way on the choice of axioms for set theory. It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.

In math a point is usually a position dependent upon a framework. Take away the framework, does the point still exist? It seems to since it crops up in framework after framework. Is a point then eternal? Is it possible to destroy a point, or would that require a point being material? :roll:

I hereby revoke my two axioms and my original frivolous statement about integers.

Happy Holidays! :nerd:

https://en.wikipedia.org/wiki/Equivalence_class

If you are using mathematical symbols, then the above should suffice. If you are arguing why a thing is the same as itself, then jump into the deep end of the metaphysical pool and splash about.

:yawn: