Mathematics is based on deductive logic and deductive logic is valid. This reduces the question of the veracity of the theorems to the original premises, these original premises being called axioms of the theory in question. Mathematical logic is the study of formal logic within mathematics. Major sub-areas include model theory, demonstration theory, set theory, and recursion theory.

Research in mathematical logic usually addresses the mathematical properties of formal logical systems, such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish the foundations of mathematics. Mathematics is about understanding patterns of all kinds, not just quantitative ones, and mathematical reasoning is based on logic. If you don't have experience with mathematical proofs, you don't know what mathematical reasoning is really about, so you won't understand why mathematics is based on logic.

Wright went on to assert that Hume's principle can be considered a logical truth. If so, then at least second-order Peano arithmetic can be reduced to logic only. Thus, a new form of logicism was born; today, this vision is known as neologism (Hale, 26%, Wright, 200). Most current mathematical philosophers doubt that Hume's principle is a logical principle.

In fact, even Wright later tried to qualify this statement. However, many mathematical philosophers consider the introduction of natural numbers through Hume's principle attractive from an ontological and epistemological point of view. Linnebo argues that, since the left side of the Hume Principle merely re-sculpts the content of the right side, it doesn't take much from the world to make Hume's Principle a reality. For this reason, he calls natural numbers and mathematical objects that can be entered in a similar way light mathematical objects (Linnebo 201).

The languages we speak in our daily lives (English, Chinese, Spanish, etc.) This means that they naturally evolved over thousands of years of human beings who tried to communicate better with each other. Mathematical logic, on the other hand, is a “formal language”. That means that their grammar and vocabulary were built intentionally by mathematicians and logicians to serve a specific purpose. Although the study of logic in the language of mathematics is relatively new, the study of logic in human languages is very old.

Instead of using the mathematical operators you're used to (plus, minus, multiplying, dividing), Boolean logic uses the operators “y” or “xor” and “no”. In short, mathematical logic tries to understand mathematical concepts through patterns that seem natural to the brain. I suppose my totally uninformed opinion is that mathematics is logical and can function as an abstract theory, but the psychological reality is that human minds don't reason with all forms of information equally, and therefore it's more useful and informative to restrict conceptual mathematics to reasoning with a certain type of information, rather than defining all forms of reasoning as mathematics. Mathematics does not focus on the symbolic manipulation of quantities, just as literature is not based on the symbolic manipulation of letters.

A mathematician who studies syntax could be working on translating a “sentence” from the language of mathematics into a set of instructions that a computer can understand. In contrast, in mathematics, finding 1,000 or 1,000,000 examples of something that works never counts as proof.