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Am I correct in stating that the study of topology is purely theoretical?

To clarify, the real world is discrete or quantized (ie; digital) whether we are discussing atoms, quarks, or strings, etc; but topology seems to depend upon everything being continuous or analog.

For example, if I draw a one inch line and a two inch line on a chalkboard, there is a topological mapping of every point on the one inch line to the two inch line, but in actual fact there are twice as many particles of chalk (or atoms, etc.) on the two inch line as there are on the one inch line. What real world value then does the study of topology serve?

Recommendations for books or publications that answer this fundamental question are welcome.

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  • $\begingroup$ Here’s Grant Sanderson aka 3Blue1Brown solving a discrete math problem using topology: youtube.com/watch?v=yuVqxCSsE7c $\endgroup$ Feb 26, 2019 at 14:08
  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Pedro
    Feb 26, 2019 at 15:53

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You have asked a philosophical question, not a mathematical one.

If by "purely theoretical" you mean something like "useless in the real world" then the answer is no. Topology has contributed significantly to our understanding of differential equations, which model many useful real world phenomena. In quantum mechanics, resolving the puzzling "wave particle duality" depends on seeing how the appropriate mathematical model can be treated with the mathematics of continuous systems. Without that kind of analysis we couldn't understand transistors and build computers. String theory (which has not yet been useful in the real world) relies on lots of topological ideas - in particular, on the study of Calabi-Yau manifolds.

When you say

the real world is discrete or quantized

you are on shaky philosophical ground. We live in the real world, but we don't know what it "is". All we have in physics are mathematical models we use to make predictions about how it behaves. At the present time the best mathematical models for its small scale behavior are quantum mechanical, but they don't say the real world is "really made up of discrete pieces".

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    $\begingroup$ And those supposedly "discrete" quantum mechanical models are actually expressed in terms of the mathematics of continuums. Perhaps it is possible to develop a truly discrete theory of quantum mechanics, but I've never heard of one. $\endgroup$ Feb 25, 2019 at 19:54
  • $\begingroup$ the tricky part to me seems to be that our computers are mostly discrete, while our theories are mostly continuous... $\endgroup$
    – don bright
    Feb 26, 2019 at 1:25
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    $\begingroup$ @PaulSinclair that's because QM is not discrete at all. It only says that certain phenomenon are discrete. Saying everything is discrete in QM is like saying everything is relative in relativity. $\endgroup$ Feb 26, 2019 at 3:06
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    $\begingroup$ @PyRulez - Those "certain phenomena" include everything we can observe. That being the case, we cannot say with complete confidence whether the continuous underpinings we have given the theory of QM are requisite, or just an artifact of our lack of imagination. So the best we can say with confidence is that our continuous models work. We cannot say that they are the only things that work. $\endgroup$ Feb 26, 2019 at 3:38
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    $\begingroup$ @JeppeStigNielsen I think the circular assertion that "the real world is the world we live in" is just about the only possible philosophical position. It works even if our world (or, better, my world) is just a simulation in some other being's computer - a science fiction trope. The only thing you have to believe is that anything exists at all. Is that shaky? (None of this has any connection to the "real numbers". There the adjective has no literal meaning - it's a historical artifact. You have to discuss the reality of mathematical objects with Plato .) $\endgroup$ Feb 26, 2019 at 13:08
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I would invite you to read Robert Ghrist's Elementary Applied Topology or check out Adams & Franzosa's Introduction to Topology: Pure and Applied to get a taste of some of the numerous applications of topology in the real world, regardless of whether the mathematical models involved match reality perfectly at all scales.

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  • $\begingroup$ Thank you. I have ordered a hardcopy of Elementary Applied Topology from Amazon and also downloaded the PDF from the author's website. I am currently reading "The Shape of Inner Space" by Shing-Tung Yau and Steve Nadis which is excellent because the authors take topology step by step from Yau's earliest beginnings in poverty, but when he gets to certain classes like Kfaler and Chern, etc., it becomes almost impossible for me to visualize. I've been retired for many years and although my undergraduate college major was physics we only touched on topology briefly, and that was many years ago. $\endgroup$
    – Ron Dotson
    Feb 26, 2019 at 23:46
  • $\begingroup$ @RonDotson: You might also like Topology Illustrated by Peter Saveliev, come to think of it. (See inperc.com/wiki/index.php?title=Topology_Illustrated for a draft version of the book.) $\endgroup$
    – J W
    Feb 27, 2019 at 17:46
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According to a report from the National Institutes of Health,

topological properties of the genetic material [...] influence virtually every major nucleic acid process.

One might be tempted to say that all life on earth depends on topology.

For more information, do a web search for "DNA topology".

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Topology is possible to apply, but to apply it you need to know it, which not particularly many engineers do. :)

Same with any other higher math. You can do really cool applied abstract algebra things too, given you get guy responsible on project to thumb you up. This rarely seems to happen if they are not confident they understand it.

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