June 30th, 2017, 11:21 AM  #11 
Senior Member Joined: Aug 2012 Posts: 1,414 Thanks: 342  Are you saying you understand advanced algebraic geometry, category theory, schemes, and all of that? Are you making that claim? Or what exactly was your reference to Grothendieck all about?

June 30th, 2017, 12:14 PM  #12 
Senior Member Joined: Dec 2015 From: Earth Posts: 153 Thanks: 21 
By math a number is value. but it can switch into measure ( example : physics)

June 30th, 2017, 12:51 PM  #13 
Senior Member Joined: Sep 2015 From: CA Posts: 1,264 Thanks: 650  
June 30th, 2017, 01:42 PM  #14 
Senior Member Joined: Aug 2012 Posts: 1,414 Thanks: 342  
June 30th, 2017, 09:44 PM  #15 
Senior Member Joined: Dec 2012 Posts: 979 Thanks: 24 
I have again to say that probably the better word I've to use is not measure, that remember the act of measuring, but "distance from". And yes here we are playing aroud the concept of exactness (my 1/K) we use to measure the area bellow a curve we know is not like polygons and parabola derivate (and is clear that when we measure areas is usefull to use the same scale for X and Y), so can't be squared with a linear measuring instrument, but just going infimus to the integral, where the concept of "unit of measure" don't loose its signfy (we fix by our definition), but where the precision of the measure is $\infty$ 
June 30th, 2017, 11:04 PM  #16 
Senior Member Joined: Dec 2012 Posts: 979 Thanks: 24 
What is already known and clear is that the precision of the measure depends on the right precision of the instrument we use, so depends on witch $K$ we choose, but a bigger $K$ is not an insurance of a better measure ! See the above integration via Step Sum of a quarter of an Ellipse:  From K=10 to K=20 the error fells lot  from K=20 to K=50 is quasi linear but:  K=50 gives a better result than K=60, and this is a very big problem for physics study... In this case 10time more precision is for sure better, but is not always an insurance... 
July 3rd, 2017, 04:57 AM  #17 
Senior Member Joined: Dec 2012 Posts: 979 Thanks: 24 
To answer to Maschke and idontknow and some other: Both concerning are not right since here you can see that the math is right, but you can't use the infnite precision you need to complete the Sum:  floating point here affect the result lot due to the high number of terms and...  if you loose a digit like 1 or 2 you loose not so much, but if you loose 8 or 9 you loose lot. and if you keep the wrong K that get most of the high digit approximation you can have a worst result than using a littlest (not soo much) K that collect most of the lower. Therefore you make good math, but you've always "wrong" results if you don't know exactly what you are trying to measure and witch instruments. This problem is well known in electric circuits, for example when you've to decide if measure Volt or Ampere in a circuit: Volt comes from high resistence in the instrument (but not infinite), Ampere from low resistence in the instrument (but not zero). I introduce the Complicate Modulus Algebra for the nth Roots: you can play just with 2 integer value and you will not loose some digit in the computation of the Sum of several roots (in theory). Problem is that is very slowly and usefull just for few problems involving powers. The very good news is that after a long period of work on the "definitions" I'm finding the first theorem on the Rest. Results are under checking... Last edited by complicatemodulus; July 3rd, 2017 at 04:59 AM. 
July 3rd, 2017, 06:21 AM  #18  
Senior Member Joined: Jun 2015 From: England Posts: 624 Thanks: 170  Quote:
I measure the number of £51 notes in my pocket as exactly zero I also measure the number of £1 coins as exactly 3.  
July 3rd, 2017, 11:09 AM  #19 
Senior Member Joined: Aug 2012 Posts: 1,414 Thanks: 342  I agree with your point. I think we need a philosopher of physics to explain why counting is exact but measurement isn't. My remark holds if we distinguish measurement from counting. For example the number of coins is exact but their weight can never be exact.
Last edited by Maschke; July 3rd, 2017 at 11:12 AM. 
July 3rd, 2017, 11:29 AM  #20 
Senior Member Joined: Jun 2015 From: England Posts: 624 Thanks: 170 
Counting is not always exact, but it is a well recognised and widely used form of measurement. My examples were artificially constructed and arguably the count or measurement is a pure number. But you find a Geiger counter measures counts per unit time When you go to hospital they measure your white cell count in counts per unit volume or (microscope slide) area Biologists, agronomists etc measure all sorts of biological counts per unit space and so on. It could be argued that the number of strokes to complete a hole or course in golf is a measure of skill, whether it counts as just a number or not is debatable. But all these types of measurement have one thing in common with other measurements  they are never 'exactly right' even though the answers are integrers. But no matter how many times I count the notes in my pocket I will never find a £51 note, although I might have differing numbers of £1 coins. Is that enough philosophy for you? If not can I refer you to Scientific Inference by Harold Jeffreys of @Mathematical Physics' fame. Both are Cambridge University Press I think 

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