Posted by Noor Salik

**Different forms of mathematical thought**

One makes the distinction in mathematics between:

(i) **Continuous thinking **(for example real numbers and limits), and

(ii) **Discrete thinking **(for example natural numbers and number theory).

Experience shows that continuous problems are often easier to treat than discrete ones.

The great successes of the continuous way of thinking are based on the notion of limits

and the theories connected with this notion (calculus, differential equations, integral

equations and the calculus of variations) with diverse applications in physics and other

natural sciences.

In contrast, number theory is the prototype for the creation of effective mathematical

methods for treating discrete problems, arising in today’s world in computer science,

optimization of discrete systems and lattice models in theoretical physics for studying

elementary particles and strings.

The epochal discovery by Max Plank in 1900 that the energy of the harmonic oscillator

is not continuous but rather discrete (quantized), led to the important mathematical

problem of generating discrete structures from continuous ones by an appropriate,

non-trivial quantization process.

**Oxford Users’ Guide to Mathematics**