[Bill S (OP)]

There’s a chance I may not continue posting, and there are a few things I would still like to clarify, just in case.  Please have patience if I dig up some “oldies”.

Looking back at past exchanges, I find myself thinking that there are areas in which we tend to go round in circles.  This may be because some of us “hitch-hikers” are slow to catch on. It may also be because experts often answer the question they think should have been asked, rather than the one that was actually asked.  Then there are crossed wires……

One point that needs to be cleared up arises from:  Can we lay nothing to rest? https://www.thenakedscientists.com/forum/index.php?topic=53002.msg445560#msg445560  Where Pete says:

But you haven't been appreciative enough to take my advice and read what I suggested. I feel as if you're being disrespectful to me when the fact is that I'm being as respectful to you as I possibly can by doing a lot of work trying to help you understand this and all you do is skirt around my one single suggestion which will help you

It saddens me that, in asking a question about something that an expert has said, I might give the impression that I am disrespecting that person’s expertise.  Such is not the case, but I will never pretend that I understand something, if I don’t. Therefore, I ask questions.

I did read the information Pete attached, it certainly improved my understanding of mathematical infinities; thanks Pete.  However, there still remain a few unanswered questions.  This may be, in part because, before we can say anything meaningful about a concept, we have to be quite clear about the context in which a question is being asked, and an answer given.

Before even attempting to go any further, I should check, and possibly improve, my understanding of the mathematical concept of infinity.  Hopefully I will post my thoughts soon.

[alancalverd]

Before even attempting to go any further, I should check, and possibly improve, my understanding of the mathematical concept of infinity.  Hopefully I will post my thoughts soon.

That path leads to insanity, as many mathematicians have shown!

However most physicists and engineers seem to die with their brains relatively intact, having decided that infinity is a tool determined by context and either means  "Limit_x→0 {A/x}" where A is pretty well anything (physics) or "too big or far away to make any practical difference" (engineering).

Your departure would diminish the challenge and value of these boards! Trying to explain something I think I understand to someone who thinks he doesn't, really gives the grey matter a good workout, and reading the opinions of experts in fields I haven't even thought of is just as much fun. 

Just as Eddington said "the student of physics must become accustomed to having his common sense violated five tims before breakfast", so anyone paddling in the shoals of science must be prepared for some fairly robust exchanges with the whales that explore the deep waters.   

[jeffreyH]

You are one of our valued members Bill. Don't ever forget that. In amongst all the madness that ensues you are a voice of reason. We have too few as it is.

[Bill S (OP)]

Alan and Jeffrey, thanks for the encouraging comments.  There are times when I feel it would be less frustrating, just to draw a line under all this, rather than keep “picking” at it; but I’m not really that sort of person, so I’ll try to keep a toe in the door.

Insanity is not a problem for me – I know I’m mad; so here are some thoughts about mathematical infinity, inviting criticism.

1. Mathematical objects do not exist in the same sense that physical objects exist.  “If a mathematician could write down a set of non-contradictory axioms and rules for deducing true statements from them, then those statements would be said to ‘exist’.”  (Barrow)

2. Infinity is not a number. It may be found used in such a way that it looks like it is being treated as a number, but this probably arises because of “shorthand” terminology. Eg. Process “A” is effectively endless, so it is referred to as “infinite”. This does not claim that there is a point, identified as “infinity” that can actually be reached.  This would be a contradiction in terms.

3. Infinity is meaningful in terms of a topological space in which two sequences of objects converge, but never meet.  Thus, the sequence 1.1, 1.01, 1.001, 1.0001, etc. converges to the number 1, but never reaches 1.  The “shorthand” way of expressing this is to say that the sequence is infinite.

4. In calculus, a sequence like 1, 2, 3, 4 is considered as "converging to infinity", but is there actually an object "infinity" that this sequence is converging towards? 

Mathematically, there is.  If you can define a topological space which has the real numbers as a subset, plus an additional object called "infinity", and if the notion of convergence is such that sequences converge to this additional object.  In that sense, infinity exists.  (I don't believe I said that )   

[geordief]

In that sense, infinity exists.  (I don't believe I said that  ) 

Does "infinity" look very small when viewed from afar (eg written as a symbol)  but  balloon  to the size of the whole universe when approached in detail?

A bit like my seemingly wrong idea of "Russian-dolled"  events in the "events" thread I  just started.... where the fall of Rome seems like one event from a distance .

[alancalverd]

Infinity exists in the same way that God exists: it is a word that we use as we wish, to convey whatever is appropriate in context.

E.g. "God knows" = there may be a rational explanation for the status quo but looking for it won't help us solve the present problem.  "Focus to infinity" = set up to converge parallel beams to the focal point.

[geordief]

Yes, it seems pedantic to have to repeat that infinity is a mathematical  process rather than a thing in its own right (or a word  that can be used loosely to convey  various useful ideas)

But it seems that  there may be a natural  inclination  to want to  make it into  or picture  it as  a real thing   when it is not.

Even so,it does seem a genuine question whether or not the universe itself  is finite or infinite  (I go for infinite-and am also happy with the "infinite" level of lack of understanding that quite probably entails)

[Bill S (OP)]

Infinity exists in the same way that God exists: it is a word that we use as we wish, to convey whatever is appropriate in context.

So, where does that leave my attempt to find a description of mathematical infinity?

[Bill S (OP)]

Infinity exists in the same way that God exists: it is a word that we use as we wish, to convey whatever is appropriate in context.

“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”

The original Humpty Dumpty may have been a cannon, but, could be, this one was a scientist.

[alancalverd]

So, where does that leave my attempt to find a description of mathematical infinity?

Try Brian Clegg's "Brief History of Infinity" - a very readable summary of most infinities, including the reason why your search is doomed!

[Bill S (OP)]

I should probably re-read that, but at this point, I was just considering my understanding, or otherwise, of the concept of mathematical infinity. One step at a time.

[geordief]

I learned that one mathematical infinity can be bigger than another. It was schoolboy stuff ,if a little above my head then.

Are there any practical applications of this finding or is it purely theoretical? (even if obviously true)

[pensador]

If you can define a topological space which has the real numbers as a subset, plus an additional object called "infinity", and if the notion of convergence is such that sequences converge to this additional object.  In that sense, infinity exists.  (I don't believe I said that )

Infinity is number that can never be attained, a distance that can never be travelled in an eternity, a time that can never be reached. ie Infinity has no limit it is unending.


Ps it is also useful in maths to simplify equations as f(a value) → ∞ something can be ignored in an equation, making the sums easier.

[Bill S (OP)]

Infinity is number that can never be attained,

Yes, we do go round in circles.  Infinity, it seems, is not a number, unless we choose to treat it as a number.

As Alan pointed out: "....it is a word that we use as we wish, to convey whatever is appropriate in context.".

[jeffreyH]

Infinity is never a number. If there is ever only one thing you learn about infinity that should be it. What is 5 times infinity? It is undefined in mathematical terms.

[Bill S (OP)]

You are preaching to the converted, there, Jeffrey.

2. Infinity is not a number.

Just one of the many times I've said it; usually in response to its being used as though it were a number.

BTW; if you have any comments about the points in #3, I would welcome them.

[pensador]

You are preaching to the converted, there, Jeffrey.

2. Infinity is not a number.

Just one of the many times I've said it; usually in response to its being used as though it were a number.

BTW; if you have any comments about the points in #3, I would welcome them.

OK

Infinity is concept of a number that can never be attained ie N/0 →∞. Your number sequence given in 3 is correctly infinite although the number 1.01, 1.001 ultimately → 1.0
"
4. In calculus, a sequence like 1, 2, 3, 4 is considered as "converging to infinity", but is there actually an object "infinity" that this sequence is converging towards?  " It is converging towards the concept of a number that can never be attained.

Infinity is a mathematically conceptual number that can never be attained or written or imagined, except to say it is bigger than anything ever imagined. ie Big

 

[pensador]

Infinity is never a number. If there is ever only one thing you learn about infinity that should be it. What is 5 times infinity? It is undefined in mathematical terms.

Maybe my mind is slipping but "undefined" would be something that is undefined in mathematics, ie 0/0 0^0 can not be defined etc N/0 →∞ is not strictly undefined, in that we know which way the number is headed, if we were to try and calculate the outcome. Although ∞/∞ would be undefined like 0/0 https://en.wikipedia.org/wiki/Undefined_(mathematics)

[Bill S (OP)]

My understanding is that “undefined”, like other terms we have been looking at, is used in a variety of ways.  This can be very convenient, and facilitate scientific progress, or it can be bl**dy frustrating, depending on your perspective.

We have been considering the nature of a point, and its connection with an event.  It surprises me that no one has “slammed the door” on this by asserting that the term “point” is undefined.

[pensador]

My understanding is that “undefined”, like other terms we have been looking at, is used in a variety of ways.  This can be very convenient, and facilitate scientific progress, or it can be bl**dy frustrating, depending on your perspective.

We have been considering the nature of a point, and its connection with an event.  It surprises me that no one has “slammed the door” on this by asserting that the term “point” is undefined.

I thought you had defined the point of the OP

The point is "a single item or detail in an extended discussion, list, or text" An event is something that occurs at a point in space time.