Arithmetic Computation

The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.
Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of double-checks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter.

Mathematics is not numerology

Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem proverbs. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Pseudo mathematics

Pseudo mathematics is a form of mathematics-like activity undertaken primarily by non-mathematicians. The word is adapted from the term pseudoscience, which is applied to ideas that purport to be scientific but are not. People who practice pseudo mathematics are sometimes called pseudo mathematicians. Pseudoscience is a term commonly applied to any body of knowledge, methodology, or practice that is portrayed as scientific but diverges substantially from the required standards for scientific work or is unsupported by sufficient scientific research.
The term "pseudoscience" appears to have originated around 1844 as a combination of the Greek root pseudo, meaning false, and the Latin scientia, meaning knowledge or a field of knowledge. It generally has negative connotations because it asserts that things so labeled are inaccurately or deceptively described as science. As such, those labeled as practicing or advocating a "pseudoscience" typically reject this classification..