Refutation of the \"Language-Only\" Interpretation of Math (The limitations of science)

by dhw, Friday, March 12, 2010, 13:53 (5149 days ago) @ xeno6696

We could probably go on indefinitely offering examples and counter-examples to show that "twoness" does or does not exist independently of human observation and language. I think my example of the snowdrop and snowflake (yes, two flowers, but no, not two snowdrops) shows pretty conclusively that "twoness" requires a human act of association, but your helium v. hydrogen v. lithium makes a good case for your version. I was inclined to leave it at that, but out of interest I googled "Do numbers exist?" and found an article by Lee Lady: www.math.hawaii.edu/~lee/exist.html. I'm afraid I haven't had time to read and digest it fully, but I noted down the following: 
 
"The prevailing opinion among mathematicians, at least as far as I know, is that mathematics has to do with a man-made universe, a mental universe, completely separate from the "real world," whatever that may be. But it takes a highly intellectually sophisticated mind to think that supernovas and electrons are real but that numbers such as 6 and 59 are not."
 
This suggests he's on your side (except that he thinks the opposition is more widespread than you do). However, it may not be so, as you will see later from another quote.
 
There's no disagreement between us on the importance of maths or its relation to our real world. Obviously physics and man-made activities such as engineering and architecture depend on it, and I'll take David's word for it that "the universe and biology are both very comprehensible through math formulas." The reason why I've challenged you is your claim that "twoness" is as real as the sun, and all mathematics trace their lineage back to the natural observation of "twoness". However, you also write: "a voice in the back of my head likes to remind me of the question "How is this question practical?" ... and a similar voice in the back of my head is asking whether it really matters whether twoness does or does not exist independently! Much more important to me is whether mathematicians are in a position to explain the mechanisms of life and the universe, as David suggests. And do their formulae imply a conscious intelligence at work (David's view), or a natural, unconscious order of things (George's view). Here is another quote from Lee Lady (but other passages in the article suggest he is not religious):
 
"I believe it was Kronecker who said, "The natural numbers were created by God; all the others are the invention of humans." I believe that most contemporary mathematicians would agree that Kronecker was wrong only in his statement about natural numbers; they too are the creation of human minds."