The problem in arithmetic is people use applications to physical phenomena without the necessary experience to understand if it even applies. Examples of arithmetic applications can be absurd. I like Grey Goose Vodka and if a quart of alcohol is mixed with a quart of water it will yield ~1.8 quarts of vodka, which is true of most mixtures of alcoholic liquids. I play basketball and if one night I hit 2 out of 3 shots and the next night I make 3 out of 4 shots and you compute my average over both games by adding the two fractions in the usual method of adding fractions, which is find the common denominator:
2/3 +3/4 = 8/12+9/12=17/12
So, I made 17 shots in 12 tries? The method of adding fractions does not give my shooting average over separate games.
A new way of adding fractions is add the numerators and add the denominators:
2/3 +3/4=5/7
Experience tells us this arithmetic method is correct.
Under normal rules 2/3=4/6 but not under this new method of adding fractions:
4/6+3/4=7/10, and 7/10 does NOT equal 5/7.
In normal arithmetic, fractions behave as integers. So consider the numerators 2 and 4. The numbers can be fractions: 2/1 and 4/1.
Now use our new arithmetic:
2/1+4/1=6/2 rather than the 6/1 obtained under normal rules.
This has a major significance since it concludes that there is no truth in mathematics but simply a tool developed by man. Only experience can show where ordinary arithmetic applies to any given physical phenomena. The theorems of mathematics are not truths but man's tool to model aspects about the physical world. As far as the study of the physical world is concerned, mathematics merely offers theories. Mathematics is based on man's intuition whereas axioms of logic or truth is an art where man is allowed to paint his better wrong answer with confidence.
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