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Truths: Mathematics and History

Truth in Mathematics

Although there does not exist a generally acceptable definition of Mathematics, it is believed to be the base of Sciences, as well as the most commonly-used tool in Science subjects. Mathematics itself is an independent system based on its truths and logic.

Rule of the Game: Axioms as Primary Truths in Mathematics

Axioms are the foundation for all mathematical theories. They are considered the primary truths in Mathematics. An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.’

Axioms are typical examples of Pluralist since they contain mixed categories of truth. Some of them are absolute and objective truths, for example, 1,2,3......are defined as natural numbers. They are somehow empirical. For instance, we can definitely prove that the sum of angles in a triangle is 180 degrees. But most of them are non-empirical as they cannot be tested.

Mathematical theories as Secondary Truths: Comparing with Scientific truths

Mathematical theories are the secondary truths in Mathematics because they are established from axioms.

Mathematical truth and scientific truth lies in the direction of the reasoning. In science, there is (we assume) some set of rules that defines 'truth'; and we try to guess what they are by observing outcomes. In math, we know what the rules are, because we get to make them up; and we then try to see what outcomes follow from those rules.

Realizing and Evaluating Truths in Maths

Axions Do Not Need to Be Proven

Although axioms are typically called 'assumptions', they are not assumptions in the same way that we mean it in our daily lives. A mathematical assumption, an axiom, is not of the same sort - it's more like a definition - and from the perspective of the mathematician, the reason that it is safe to make this sort of mathematical assumption is that as a definition, it cannot be wrong. Now if axioms 'can not be wrong', this sounds like a promising possibility - it might even suggest we have a pathway to truth .

Deduction

Deduction is the main way how mathematical theories form and proceed.

Dependency and Independency: Conclusion and Evaluation

Axioms are the foundation of the subsequent and further study of Mathematics. The non-logical ones is simply a formal logical expression used in deduction to build a mathematical theory. (e.g. parallel postulate in Euclidean geometry).

Truths in mathematics are'formal', that is to say, self-contained. No mathematical truth exists independently of a set of axioms and a set of agreed-upon rules for deducing theorems from axioms. They all come together as a system; and strictly speaking, they have no necessary connection to anything outside the system.

Hence G.H Hardy's claim about the reality of Mathematics:

"A mathematician is working with his own mathematical reality. Of this reality, I take a “realistic’ and not an “idealistic” view. This realistic view is much more plausible of mathematical than of physical reality because mathematical objects are so much more what they seem. 2 and 317 have nothing to do with sensation, and its properties stand out more clearly the more closely we scrutinize it.

Pure mathematics, on the other hand, seems to be a rock on which all idealism flounders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so because mathematical reality is built that way. "

Maths : discovered or invented?

Another interesting aspect in Mathematics is the debate on whether it is inherently present in nature and thus has been discovered by humans, or if it is a human-made construct itself, that we use to understand the world around us. According to Platonic theory, mathematics exists independently in space and time. We, as humans, are trying to solve the mysteries of the universe through an attempt to uncover the secrets of mathematics. Therefore, the truths in mathematics would have to be absolute and correspondent, in line with what is observed in nature. In contrast, others believe that we are merely devising tools which model the structure of our universe to a close approximation (for example the model of elliptical planetary orbits, which is not an exact science). As a result, mathematics in this case is constructivist in nature as well as analytic, being a concept or a tool that aids us in comprehending reality but is not reality itself. Thus, depending on the angle with which one approaches the field of mathematics, truth can have surprisingly different and opposing meanings and connotations.

Truth in History

History, as an academic discipline, is the study of past events; of the recorded past. The recorded past is studied using the concepts of evidence, reliability, causation, interpretation and accuracy.

There are indisputable, objective truths in the discipline, that can be corroborated against sources, such as dates of birth of historical figures. However, this only applies to basic facts.

While History aims for objectivity, there is a general consensus that there are hardly any absolute truths in the discipline, only relative. This is because the past is gone; historians are unable to observe the past directly, and can only make historical judgments— claims made based on an interpretation of the evidence that constructs a narrative. Thus the knowledge produced is often incomplete and inaccurate.

To present an extreme view on the aforementioned; if evidence of an event in history is not taken from its original source, but from a secondary source or chain of communications, it must be presumed false. Furthermore, if the fact that an event occurred is not doubted or scrutinised for an extended period of time, it has a greater probability of being false. More abstractly, it is claimed that an event did not occur, by modern definition, if it happened before recorded history began. This idea relates to the philosophical argument 'if a tree falls in a forest', the idea that an event cannot be proven unless it is observed and recorded by a reputable witness.

Evaluating Truths: Historical Method

The Historical method is the collection of techniques and guidelines that historians utilize in their research and then writing of the past. Historians study a wide range of sources, both primary and secondary.

Primary Sources

Primary sources provide a first-hand account of events, often created at the time at which it occurred. These include oral or written testimonies such as diaries, interviews, and speeches. However, they can also include sources that are created later, such as autobiographies and creative artworks. As primary sources reflect the perspectives, emotions, and memory of a participant or observer, they are inherently subject to inaccuracy and bias.

Secondary Sources

Secondary sources attempt to provide a description or explanation of primary sources through analysis, interpretation and evaluation of them. Examples of these include journal articles, textbooks and opinion pieces. Similar to primary sources, many secondary sources are also subjective and contain bias.

How important truths are: Reliability-- Source Criticism

Evaluating sources, whether primary or secondary, is a key part of the rigorous research process. Historians read sources with a critical eye— they question the nature, origin, content and purpose of a source; and cross-check them against other evidence and sources, to obtain a more balanced understanding of the event.

Procedure for Source Criticism

Bernheim (1889) and Langlois & Seignobos (1898) proposed a seven-step procedure for source criticism in history.

1. If the sources all agree about an event, historians can consider the event proved.
2. However, the majority does not rule; even if most sources relate events in one way, that version will not prevail unless it passes the test of critical textual analysis.
3. The source whose account can be confirmed by reference to outside authorities in some of its parts can be trusted in its entirety if it is impossible similarly to confirm the entire text.
4. When two sources disagree on a particular point, the historian will prefer the source with most "authority"—that is the source created by the expert or by the eyewitness.
5. Eyewitnesses are, in general, to be preferred especially in circumstances where the ordinary observer could have accurately reported what transpired and, more specifically, when they deal with facts known by most contemporaries.
6. If two independently created sources agree on a matter, the reliability of each is measurably enhanced.
7. When two sources disagree and there are no other means of evaluation, then historians take the source which seems to accord best with common sense.

Comparison

How Truth is Perceived in History and Mathematics

Whilst truth within History and Mathematics can be perceived in very different ways, with History as subjective and Mathematics as empirical, they are considered similar in that they are both partial to scrutiny in their accuracy. Both disciplines can be considered abstract in that their origins are often unclear or a truth is not universally agreed.

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