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Is mathematical knowledge certain knowledge? Does it apply to all knowledge areas?
If the universe is mathematically ordered, this would explain the utility of mathematics but not how mathematical knowledge is acquired. Mathematics expresses the principles of this order and gives insight into it. The path to mathematical knowledge is linked to its utility, through the idea that the senses reveal to us only the changing phenomenal world of sense experiences. Mathematics is universal and constant, so to garner knowledge about principles underlying the order of the world in which we live we have to turn from sense experience and recognize the ability of the intellect to understand the ordered nature of the universe. Mathematics is perhaps the study in which there seems to appear a strong use of the word "knowledge," because information must be certain, or very close to certain, before it can be known. For this reason, it requires a stronger degree of justification then other areas of knowledge. This all stems from the belief that mathematics is the only truly constant area of knowledge; mathematical principles remain the same in all situations, with few exceptions. In formalism, mathematics is nothing more than a game, employing the manipulation of symbols that represent nothing. Formalists hence believe that mathematics conveys no certain knowledge, and is nothing but the organization of symbols. Constructivism, an opposing view, represents the idea that mathematical knowledge can be acquired by mental processes. Math therefore exists solely in the mind of the mathematician and is indubitably true. This view, consequently, does not take into consideration the application of mathematics in the "real" world, and the ability for people to simultaneously recognize mathematical patterns in nature. Depending on which view one holds, mathematical knowledge can be viewed as both certain and uncertain. Many of the areas of knowledge rely on logical analysis to make knowledge claims. Mathematics also follows suit. It is the repeating accuracy of mathematical knowledge claims that allows us to place mathematics on a immutable pedestal of unerring knowledge. And because the world is ordered around this mathematical knowledge, it can be applied in all of the areas of knowledge (aside from the hard sciences such as chemistry); however, the claims become less specific and more encompassing. The mathematics associated with sociology, psychology, and economics is much less exact. Human behavior encompasses such a wide spectrum of activity that it would take an immense effort to quantify all possible behaviors. At present, one can only paint in broad terms simple relationships that appear to exist between mathematics and human behavior. Still, there are some tangible relationships that can be established.
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