The concept of infinity in mathematics and science

[RobertJS]

How can one answer that to your satisfaction?

Don't think of me, think of satisfying reason and logic. The answer is either Yes or No, in reality.

[Geremia]

"Not speculatively, but in reality, when heading for a destination point, either there is endless further cutting a half of the remaining distance, or there is not."

So, your question is really about Zeno's "paradox"?

Perhaps this will help, from Dr. Romero Carrasquillo's translation of Fr. Hugon, O.P.'s Cosmology:

IX. – CONCLUSION: "Quantity in itself, that is, considered mathematically, is divisible ad infinitum; but physically, that is, as it is found in physical things, it cannot be divided ad infinitum."
   Proof of the First Part. Whatever has an infinite number of proportional parts is divisible ad infinitum. But mathematical quantity has an infinite number of proportional parts. Therefore, it is divisible ad infinitum. The major is evident on its own terms, and the minor is clear from the preceding. Proportional parts are halves, and the halves of the halves. But to any half one can always add another half, and so on ad infinitum. Therefore. Another proof. As long as something remains extended, it is divisible at least in itself, since it retains separable parts. But a mathematical quantity, however small it may be, is essentially extended. Therefore, mathematical quantity is always indefinitely divisible. Proof of the Minor. A quantity, however small it may be, retains its ratio of quantity. But the ratio of quantity denotes extension. Therefore, a quantity, however small it may be, is essentially extended and can never be simple. Consequently, it is divisible ad infinitum.
   Proof of the Second Part. A quantity, taken physically and concretely, is measured according to its partes aliquotae.?¹ But the partes aliquotae are finite, such that at some point one reaches the last one, as was said. Therefore, a quantity, taken physically and concretely, cannot be divided ad infinitum.
   Further, a continuous body, as it exists concretely, has a determinate form and its own activity. But a form, in order to exist, requires a determinate quantity; unless this quantity is preserved, the thing will cease to be. Also the activity of bodies requires a determinate quantity; if this quantity is removed, the thing will not be able to exercise its own power, but will be broken down through the excess of the other and will be converted into something else. Therefore, the continuum as it exists physically and concretely cannot be divided infinitely and indeterminately.?²

?¹Partes aliquotae (literally, 'several parts') are repeated parts, so-called because they are repeated several times within the whole. – The Translator.

?²ST. THOMAS, In De sensu, lect. 15: "A mathematical body is divisible ad infinitum, since in it we consider only the ratio of quantity, in which there is nothing that is contrary to infinite division. But a natural body, which is considered under a form, cannot be divided ad infinitum, because, when it is reduced to a minimum, it is immediately converted into something else due to the weakness of its power" (Corpus mathematicum est divisibile in infinitum, in quo consideratur sola ratio quantitatis in qua nihil est repugnans divisioni infinitae. Sed corpus naturale, quod consideratur sub tota forma, non potest in infinitum dividi, quia, quando iam ad minimum deducitur, statim propter debilitatem virtutis convertitur in aliud).

[God_created_cats]

How can one answer that to your satisfaction?

Don't think of me, think of satisfying reason and logic. The answer is either Yes or No, in reality.

What reality?  The one of the mathematical model, or the one of the physical object desctribed
by the mathematical model?  There is a good measure of reality in both, but as Thomas Aquinas
noted, the answer is different.

Don't take me wrong, I believe in the necessity of a sound philosophical basis for scientific
investigation.  But can you imagine how we would live if physics was stalled in the state it
had reached in the XIIIth century?  We would certainly have no forum like suscipedomine.com!
Best regards.

[RobertJS]

"Not speculatively, but in reality, when heading for a destination point, either there is endless further cutting a half of the remaining distance, or there is not."

So, your question is really about Zeno's "paradox"?

Perhaps this will help, from Dr. Romero Carrasquillo's translation of Fr. Hugon, O.P.'s Cosmology:

IX. – CONCLUSION: "Quantity in itself, that is, considered mathematically, is divisible ad infinitum; but physically, that is, as it is found in physical things, it cannot be divided ad infinitum."
   Proof of the First Part. Whatever has an infinite number of proportional parts is divisible ad infinitum. But mathematical quantity has an infinite number of proportional parts. Therefore, it is divisible ad infinitum. The major is evident on its own terms, and the minor is clear from the preceding. Proportional parts are halves, and the halves of the halves. But to any half one can always add another half, and so on ad infinitum. Therefore. Another proof. As long as something remains extended, it is divisible at least in itself, since it retains separable parts. But a mathematical quantity, however small it may be, is essentially extended. Therefore, mathematical quantity is always indefinitely divisible. Proof of the Minor. A quantity, however small it may be, retains its ratio of quantity. But the ratio of quantity denotes extension. Therefore, a quantity, however small it may be, is essentially extended and can never be simple. Consequently, it is divisible ad infinitum.
   Proof of the Second Part. A quantity, taken physically and concretely, is measured according to its partes aliquotae.?¹ But the partes aliquotae are finite, such that at some point one reaches the last one, as was said. Therefore, a quantity, taken physically and concretely, cannot be divided ad infinitum.
   Further, a continuous body, as it exists concretely, has a determinate form and its own activity. But a form, in order to exist, requires a determinate quantity; unless this quantity is preserved, the thing will cease to be. Also the activity of bodies requires a determinate quantity; if this quantity is removed, the thing will not be able to exercise its own power, but will be broken down through the excess of the other and will be converted into something else. Therefore, the continuum as it exists physically and concretely cannot be divided infinitely and indeterminately.?²

?¹Partes aliquotae (literally, 'several parts') are repeated parts, so-called because they are repeated several times within the whole. – The Translator.

?²ST. THOMAS, In De sensu, lect. 15: "A mathematical body is divisible ad infinitum, since in it we consider only the ratio of quantity, in which there is nothing that is contrary to infinite division. But a natural body, which is considered under a form, cannot be divided ad infinitum, because, when it is reduced to a minimum, it is immediately converted into something else due to the weakness of its power" (Corpus mathematicum est divisibile in infinitum, in quo consideratur sola ratio quantitatis in qua nihil est repugnans divisioni infinitae. Sed corpus naturale, quod consideratur sub tota forma, non potest in infinitum dividi, quia, quando iam ad minimum deducitur, statim propter debilitatem virtutis convertitur in aliud).

Yes, Zeno's Paradox pertains to this.

The other quotes are speaking of the mathematical and the physical. But space is not physical.

The quotes, however, do serve to show that mathematics can say one thing, and reality another. In other words, reality always potentially trumps mathematics.

[God_created_cats]

Space is not physical???

How can you say that when empty space has well established physical properties, such as electrical
permittivity and magnetic permeability?  Thomas Aquinas may not have known about that, but that's
no excuse for you.

[RobertJS]

Space is not physical???

How can you say that when empty space has well established physical properties, such as electrical
permittivity and magnetic permeability?  Thomas Aquinas may not have known about that, but that's
no excuse for you.

My point is, the quotes didn't intend on considering space, just physical matter.

My excuse is that neither the Church, nor any law, requires me to have learned about those particular things you just mentioned.

And, what is your excuse for not answering my question Yes or No after repeatedly being asked?

[God_created_cats]

Look Robert, I honestly think your question is ill-posed and tricky, and as I have stated before the answer
is Yes from some legitimate viewpoint, and simultaneously No from another legitimate viewpoint.  I think
Aristotle could understand that.

Regarding the properties of space, if you're so unwilling to learn, why waste your time in the 'natural sciences'
section of SD?

[RobertJS]

Look Robert, I honestly think your question is ill-posed and tricky, and as I have stated before the answer
is Yes from some legitimate viewpoint, and simultaneously No from another legitimate viewpoint.  I think
Aristotle could understand that.

Regarding the properties of space, if you're so unwilling to learn, why waste your time in the 'natural sciences'
section of SD?

It is neither tricky, nor ill-posed. It is clinching simplicity. The principle of contradiction forbids both a simultaneous Yes AND No answer to that question.

Who said I am unwilling to learn?  I neither affirmed nor denied the two terms you presented, and I didn't need to because the excerpted quotes did not consider them. It is unreasonable for me to accept you as an authority on the two terms you presented, particularly because I am experiencing how you handle my simple question.

[God_created_cats]

Hello again,

You are entitled to your opinion as to the clarity of your question, but I beg to disagree on that.  As to my
authority, I hold a Ph.D. degree in electrical engineering from Laval University, I teach electromagnetic theory
and microwave engineering for a living, and I have designed a number of successful microwave devices
applying that theory.  I can assure you that any college physics student is familiar with the two quantities
I mentioned.  Do yourself a favor, pick a good physics textbook (Feynman's Lectures on Physics for instance),
and start reading about this wonderful universe that God has created.  You may not be required to learn
about that, but what's life if you only do mandatory things?  And don't blame me for recommending a
book by an atheist Jew, it's the truth about physics that matters in this case.

The link between physics and mathematics is somewhat mysterious (especially to the practitioner of
theoretical or applied science), but it is probably closer than most people think.  Therefore, opposing
reality and mathematics sounds fairly absurd to me.  Bye for now.

[Kaesekopf]

Not speculatively, but in reality, when heading for a destination point, either there is endless further cutting a half of the remaining distance, or there is not. It  is either Yes or No. If "Yes", then the destination will never be reached.

What is the point of this? 

Practically speaking, there is not endless "cutting a half" of the remaining distance.  If I swing my arm to punch you in the face, I will punch you in the face (so long as my aim is good and you don't dodge).

Can you measure the distance by cutting the measurement steps/units in half?  Sure, but you'll rapidly be unable to measure such small quantities.

[RobertJS]

Hello again,

You are entitled to your opinion as to the clarity of your question, but I beg to disagree on that.  As to my
authority, I hold a Ph.D. degree in electrical engineering from Laval University, I teach electromagnetic theory
and microwave engineering for a living, and I have designed a number of successful microwave devices
applying that theory.  I can assure you that any college physics student is familiar with the two quantities
I mentioned.  Do yourself a favor, pick a good physics textbook (Feynman's Lectures on Physics for instance),
and start reading about this wonderful universe that God has created.  You may not be required to learn
about that, but what's life if you only do mandatory things?  And don't blame me for recommending a
book by an atheist Jew, it's the truth about physics that matters in this case.

The link between physics and mathematics is somewhat mysterious (especially to the practitioner of
theoretical or applied science), but it is probably closer than most people think.  Therefore, opposing
reality and mathematics sounds fairly absurd to me.  Bye for now.

You have not explained in detail what you mean about it being simultaneously Yes and No. I mean, if you are going to say something, you might as well explain yourself to try to be convincing. Merely dropping a couple of scientific terms really says nothing. I mean, I know space is permeated by anything that moves through it, and permits anything to go through it.

Sorry to say, but your Ph.D. doesn't give me a cozy feeling. It means more that you know a lot of facts and formulas, but it does not guarantee logical thought, especially pertaining to philosophy. Look how you just made a logical leap in the wrong direction in a commonplace matter with your question, "what's life if you only do mandatory things?". What I said didn't logically imply that I only learn mandatory things. My experience is that people who get Ph.D. get greatly blinded to their limitations. Even looking how you handle the concept of "infinity" makes me concerned. It means never stopping, yet somehow you seem to think it does.

Where are all the brilliant men with doctorates in chemistry and physics since 9/11? They should have been gathering in strong groups to protest that the WTC7 building was demolished by carefully placed internal explosives throughout the building.  They should have protested the scam, or the security breach to a building housing the CIA! It seems to have gone right over their heads.

Anyway, the big cheese seems to have easily grasped my question.

[God_created_cats]

Allright, I certainly need to study more philosophy, but your lack of basic scientific culture is
obvious, and leads you for instance to grant respectability to the crazy ideas that have been
circulating for 12 years regarding the 9/11 events.  There was no controlled demolition that
day in New York, only two commercial aircraft intentionally crashed into the WTC towers.
The ensuing fire weakened building number 7, which eventually collapsed.  Thank you for
providing me a reminder of these historical facts.  May the Lord grant me the courage never
to waste time again reading this web site.  Farewell

[Kaesekopf]

Anyway, the big cheese seems to have easily grasped my question.

Naturally!   

[Geremia]

The other quotes are speaking of the mathematical and the physical. But space is not physical.

So, place and distance aren't either?

The quotes, however, do serve to show that mathematics can say one thing, and reality another. In other words, reality always potentially trumps mathematics.

That's because mathematics is discovered from extra-mental reality. It's not imposed on the world as the Kantians et al. think. Boethius classic definition is that "Mathematics does not deal with motion and is not abstract, for it investigates forms of bodies apart from matter, and therefore apart from movement, which forms being connected with matter cannot really be separated from bodies." (Boethius's De Trinitate).

[Geremia]

How can you say that when empty space has well established physical properties, such as electrical
permittivity and magnetic permeability?  Thomas Aquinas may not have known about that, but that's
no excuse for you.

That's not space. That's "aether" or "plenum."