Math and logic

  • Thread starter alvin51015
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In the late 1980's I asked my logic professor if there was some kind of logical and/or mathematical process which unified numerically based mathematics with true-false based symbolic logic.He told me that someone had written a lengthy book which apparently proved that it was totally impossible to do such a thing.But I keep thinking that there must be a way.So my question is whether any progress had been made in this area in the last twenty plus years.
 

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  • #3
HallsofIvy
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I really have no clear idea what you mean by "unified numerically based mathematics with true-false based symbolic logic". There were, in the late 19th century, attempts to reduce all forms of mathematics to an "axiom based" form of logic but Curt Goedel, in the early twentieth century showed that such a thing was impossible: given any set of axioms, there exist a statement that can neither be proved nor disproved from those axioms.

Indeed, rather than "true-false based symbolic logic", much of the recent work in logic has been the other way- "multi-valued logics" and "fuzzy logic" where statement are NOT just "true or false" but may have varying degrees of "trueness".
 
  • #4
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given any set of axioms, there exist a statement that can neither be proved nor disproved from those axioms.
Any set of axioms powerful enough to do arithmetic. Proposition logic is an exception, for example.

The numerically-based math reduced to symbolic logic sounds kind of like the construction of the real numbers and all that from set theory. It's not really reduced to symbolic logic, but it's reduced to sets.

As far as recent developments go, this came to mind:

http://blogs.scientificamerican.com/guest-blog/2013/10/01/voevodskys-mathematical-revolution/
 
  • #5
HallsofIvy
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Any set of axioms powerful enough to do arithmetic. Proposition logic is an exception, for example.
Yes, I should have said that.
 

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