Logic in Mathematics and Ethics

If Ethics and Maths both employ logical thinking and argumentation to produce knowledge, can claims in Ethics be as well justified as those in Math?

Before discussing the prompt, it is necessary to address the vagueness of the terms “logical thinking” and “argumentation”, and distinguish them from the formal logic used in Mathematics. While the ethics use a form of logic, it is generally either dialectic or philosophical, seeking to establish well-founded arguments or looking at the behaviour of language in relation to logic. Another question that raised itself is that of whether all forms of ethics employ formal logic, and whether logical thinking is applicable to all areas of ethics. Keeping this in mind, two (overlapping) lines of argument were formulated: the character of premises in the two areas, and the truth values of claims in Mathematics and Ethics.

A key distinction between the two fields is that the premises upon which they are based, their axioms, are different in character. While axioms in Mathematics are internally referential and, if discussing external phenomena, quantitative, those in Ethics can be linked to the Human Sciences and thus — of a certain subjectivity. While in Mathematics, axioms may be invented in response to inadequacies in existing theory (for instance, the invention of the Zermelo-Fraenkel axioms in order to resolve Russell’s paradox), such inventions hold different significance in Ethics. Perhaps a key difference between the two Areas of Knowledge is their function: while Mathematics exists primarily in a theoretical realm (though it has significant practical applications), Ethics hold different connotations as they seek to essentialise truths related to the nature of human behaviour. The reconceptualisation of axioms or premises in Ethics is thus more linked to individual perception and can either represent or bring about significant shifts in societal consciousness. Ethics prior to and after the popularisation of humanism can be linked to shifts in major value systems, and the development of Ethics as a field must be considered in the context of to the ideas prominent at a given time. While in certain times (that of the Ancient Greeks first comes to mind) Mathematics might hold similar significance, the effects of changes in axioms are arguably less marked, and more restricted within the field itself.

Furthermore, the logic employed in Mathematics generally relies on universally accepted definitions (after a certain point) which have defined ‘truth values’. It is, however, less clear if ethical statements can have the same characteristics. While formal logic may be employed in the discussion of ethical statements, it does not apply to Ethics with the same rigidity. The primary benefit of logic is argumentative rigour—that if the premises of an argument is true, then the conclusion is true as well. Such a function has its roots in classical philosophers like Aristotle, who employed logic to ensure that his arguments were sound. That being said, logical methods do not discriminate between true and untrue premises, which are the true areas of contention in Ethics. Different approaches to implementing opposing premises could be employed to reach different conclusions in Ethics, and all would be internally sound but still untrue.

Engaging with the prompt, however, it must be reiterated that it looks for “justification” and not “truth”. Argumentative logic allows claims in Ethics to be proven as internally sound, thus making them justified to that extent. However, the premises used to arrive at said claims may not be justified through this logic. In making statements on the nature of human behaviour, Ethics are vulnerable to a subjectivity in the perception of human nature, and logical thinking is ill-equipped to resolve those shortfalls.

Human Sciences #3

1. The production of knowledge in Human Science is too riddled with issues to give any credence to the claims of experts. Do you agree?
The issue with knowledge in the Human Sciences is the variability of human behaviour. In order to reduce the inaccuracy of claims, investigators can focus the scopes of their research to encompass only a specific group. In reducing sample size, it is possible to make more accurate deductions from a less variable pool of data. This is also a limitation, however, because the sample size restricts the applicability of the conclusions drawn from an investigation.
Another method that was used in our own investigations in the Human Sciences was primarily using quantitative data. More easily measured and thus somewhat less subject to individual bias, qualitative data aids the replicability and accuracy of collected data. For instance, through enumerating hours of sleep, one can gain a sense of the sleeping habits of IB students. The limitation of this tactic is clear, however, as the human sciences require qualitative data in order to accurately represent human behaviour. Human behaviour is qualitative in nature, and cannot be summarised in only numbers. Using the analogy of the sleeping habits of IB students, while the number of hours might offer some insight, they could not encompass the quality of sleep (an inherently qualitative value) experienced by the students. However, qualitative data is not necessarily unreliable. Though in its worst form subject to the researcher’s interpretation, qualitative data can be made reliable by clearly defining terms of an indicator, and using precise descriptions to minimise miscomprehension.
In a field like Psychology, which relies heavily on qualitative data, accurate experiments can still be carried out. By creating and enforcing strong standards of quality in data collection, participating in group verification, and ensuring that the data collection is as accurate as possible, psychological studies can offer valuable insight into the human mind.

2. Summarize in your own words the reading ‘Can we use scientific approach with humans?’
The article discusses commonly raised differences between the production of knowledge in the Human and Natural Sciences and evaluates them. Key points raised include the idea of ‘degrees of certainty’ (that is, the methods of data collection in some areas of the NS allow it to be more precise than in the HS) as well as less accurate areas of the NS that often aren’t discussed. In highlighting these, the author draws connections between the two areas and highlights how criticisms of the Human Sciences can often be applied to the Natural Sciences. However, the author ultimately concludes that, beyond overarching conclusions of the Human Sciences, there’s a dignity inalienable from the human condition that cannot be encompassed by individual claims.

Comparing Human and Natural Science

The Human Sciences are defined (quite loosely) as dealing with humans and their actions, as opposed to the natural world.

Although they often overlap in the phenomena they study, the Human and Natural Sciences are distinguished by the approaches they take to looking at evidence (their foci) and the methods by which they collect data. As previously visited in our study of TOK, the Natural Sciences rely on inductive reasoning, primarily through the scientific method. However, the irrational nature of humanity means that reasoned results aren’t entirely reliable. Through collecting data using surveys, —, and quantitative data, human scientists draw conclusions about the habits of people.

1 / THE VOLCANO: QUESTIONS IN THE NATURAL AND HUMAN SCIENCES

The task is to look at the photograph and propose two investigations from the Human and Natural Sciences respectively.

An interesting point about this photograph, in particular, is that it belies many geographical questions—in our study of IB Geography, it becomes clear that the field is, in fact, a marriage of the Natural and Human Sciences. Looking at space or, more specifically, the space of our world, Geography takes both physical and human factors into account in its production of knowledge. 

Human Sciences
To what extent does proximity to natural disaster influence settlement patterns?

This question could be investigated by surveying the different kinds of settlements in relation to distance from the volcano. The distribution of land use could be considered (for instance, how the distance of retail locations differs from that of residences) as well as socioeconomic disparities. The data could also be compared to similar findings in other areas, as well as existing theories on settlement patterns and land use, to draw a conclusion of how the volcano has impacted the settlement.

Natural Sciences
How do natural disasters influence the chemical composition of the atmosphere?

This question can be investigated quantitatively by measuring the concentrations of different gases in areas of varying distance from a natural disaster. Although the variables cannot necessarily be controlled, the results can be compared with chemical compositions of other areas to form a conclusive result.  

2 / MY RESEARCH TASK

I chose to create an investigation examining the links between extraversion/introversion and levels of stress. In order to reach an acceptable accuracy, the questions are either enumerated or clarified in specific terms. In my investigation, I recorded age groups and whether or not the sociability was externally verified or self-reported in order to get a sense of the investigation’s demography. In order to produce data of greater value, I identified a number of stressors and symptoms of stress. I also chose to include space for additional remarks in order to collect more data.

Exploring Correspondence and Coherence

Read the TOK essay and suggest how the student could have used the concepts of coherence and correspondence

This essay is ultimately a discussion of the significance of internal and external factors in different Areas of Knowledge (AOKs), making an argument for the practical applications of each field. Discussing the fallibilities of certainty in the Natural Science, the author briefly identifies a typology of knowledge that allows for certainty to be achieved.  The comparison made to Ethics was lacklustre at best and seemed characteristically flawed, though less relevant to this reflection.  In fact, coherence and correspondence are at the heart of this essay, which focuses on the context of knowledge production and its subsequent application. Knowledge does not exist in a vacuum, as it’s said, except that in some cases it does. The primary failure of this essay is the inability to connect it to larger concepts within TOK and epistemology—the thoughts the student sought to articulate could have been far clearer discussed when referencing established ideas. The essay as a whole (but specifically the examples used) seemed to simply be a less nuanced explanation of correspondence and coherence. 

Proofs in Sciences and the Mathematics

Proof: Defined as a(n) ” [piece of] evidence or argument establishing a fact or truth”.

The Natural Sciences consider proof in the form of empirical evidence. As they rely on inductive reasoning, proof refers to evidence used to support or refute a hypothesis. The hypothesis then determines the nature of evidence collected as well as the function of the proof. Furthermore, the key to the development of the natural sciences is the progression of measurement or observational tools in the collection of evidence. Here lies a key limitation of scientific proof: it is wholly subject to the constant evolution of its field. The reliance of key theories in science on variable evidence might make the field volatile, if not for the stringent practice of testing hypotheses, at least in ideal conditions. The use of the scientific method as a subject-wide standard by far relieves these possibilities. Thus, from these considerations, scientific proof can be found to rely on hypotheses and natural progression of the field.

Key to discussion of mathematical knowledge is recognition of abstraction of the field. In Mathematics, the term “proof”  refers to the use of previously established rules (axioms, and their derived theorems) to prove an idea. Though a discussion of axioms requires further thought, it is important to acknowledge that the premise of an axiom is that it is indubitably true. This establishes the nature of mathematical knowledge as wholly deductive—and proofs as regulators of internal consistency. All mathematical knowledge thus originates from these primary axioms, and proofs function as means of verifying derived theories.

In these two analyses, the key difference between the idea of “proof” in the two Areas of Knowledge reflect larger characteristics of the field. This question of the meaning of “proof” in fact highlights a characteristic of subject-specific language—that language may have entirely different implications in different contexts. The natures of scientific and mathematical proof highlight the differing priorities of the field: the prior seeks to reflect truths of the external world while the former questions and develops itself. Considering the idea of “proof” itself, we arrive at the following conclusion: scientific proof requires confirmation from the outside world and mathematical proof is meant to ensure that theorems are internally consistent. Though not explored in this reflection, another question that must be considered is the methods by which we arrive at knowledge in the two areas. 

From this question, the importance of semantics and considering the language we use becomes clear: in the case of the Natural Sciences and Mathematics, simple definitions can illuminate the difference in usage and connotation. But as the subtleties of language grow less clear, analytic discussion of words and their meanings are necessary to facilitate communication. We search for interdisciplinary applications of knowledge, but that cannot be accomplished when key elements of these applications, that is, the language used, is not understood clearly. From this discussion, a clearer understanding of the two Areas of Knowledge has been reached, however, the more pressing question of how to facilitate understanding between disciplines has yet to be answered.

Introduction to Mathematics

1. What is the difference between a conjecture and a theorem? A conjecture is an idea that has not been proven, while a theorem is one that has been verified according to mathematical axioms.

2. Eduardo Saenz de Cabezon uses the example of people being surprised that folding a normal piece of paper 50 times, will reach a thickness as high as the sun. He challenges us to ‘do the math’ and see that he is correct. What do you think meant when he said that Maths dominates intuition and tames creativity? Do you agree with this?
When Cabezon refers to this, he is discussing the realm of knowledge in which Mathematics exists. Through highlighting this point, he makes a statement on both the abstract nature of Mathematics and the way the human mind interacts with such abstractions. It is a curious thought to raise, frankly, characterising the sciences as more “human” than the theory of mathematics. Essentially, he purports that it is groundedness, visible consequences, that aids us in comprehending the Natural Sciences. It’s this comprehension that Cabezon labels as intuition—the arrival to conclusions in a process relying on our personal impressions of the subject at hand. This can be contrasted with the pure reason, a cold logic, that forms the basis of Mathematics. In addition to simply marking a difference between the fields, Cabezon’s comment offers insight into knowledge production in Mathematics and our own relationships with it as an Area of Knowledge (AOK).

However, I find myself unable to wholly agree with the position he highlights. Up to a certain point, I find that layman’s Mathematics (that is, mathematics less abstract than pure theory) can be quite intuitive. Basic arithmetic and many key geometrical patterns that form the basis of more complex theorems lend themselves to the development of a frame of reference analogous to that of the Natural Sciences. Once formed, this allows for a clear comprehension of Mathematics without the applications of more convoluted logic. This isn’t to say that these claims don’t need to be tested—however, the verification of knowledge claims is significant in all Areas of Knowledge. In many ways, Mathematics is more conducive to intuitive learning through pattern recognition because one has the assurance that an answer of some sort can always be reached.

3. Saenz de Cabezon claims that the truths in maths are eternal. Do you think this gives maths a privileged position in TOK?”
Though I would argue against the description of this position as “privileged”, such a reality does offer concerns vastly different from those of other Areas of Knowledge. While other fields require the consideration of falsifiability and inaccuracy, the primary concern of Mathematics can lay in the methodologies of logic and reason used to arrive at conclusions, and the fallacies of these. That isn’t to say that Mathematics does not face implications of practical limitations—it is important to understand that the field of Mathematics, like all others, does not exist in a vacuum, despite its abstractions. Applications of Mathematics, though differing from those of other fields, face their own limitations. Knowledge production in this field is perhaps more volatile than in others precisely because of that—the verification of knowledge takes place on both an abstract and a practical plane. Besides, if knowledge in Mathematics is eternal, a question stands to follow: without any change in its foundational tenets, how much of Mathematics can we really learn?

Truth in the Arts

1. Summarize what the main point(s) of the two essays into one short paragraph each.
The author of “The Truth, the Whole Truth, and Nothing but the Truth” highlights the power of art, in the form of literature, in depicting truths of the human condition both accessibly and accurately. Beyond this, however, they highlight the ability of other art forms to offer candid portrayals of the truth in a manner unachievable linguistically. Highlighting truth as a conveyance of the experiences of the human condition, they highlight the significance of emotional knowledge, often downplayed in favour of the factual regurgitation of the Human and Natural Sciences.
In “Art and Truth”, the author discusses both the kinds of truth that can be found within the arts and the modern impulse to search for truth statements in all experiences. Following a variety of artworks and the factual and emotional truths that may be extrapolated from their content, they ultimately conclude that the value of art does not have to be encompassed by learning or truth statements. This is of particular interest to me as the impressions and “sense of being moved” that are mentioned seem in fact to be another form of knowledge, and one worth exploring in greater depth.

2. How do both of these essays reflect what is presented in chapter reading about truth in art?
The chapter discusses the idea that a piece of artwork is defined by its nature as art, and not its content. In emphasising this, the author highlights a key point of consideration in the arts: that realism is not the definitive object. Instead, artists have their own rationale for their work, an artistic intention, that underlies their creations. This is important because, as reads the quote by Iris Murdoch referenced in the chapter, the beauty of artwork allows it to transcend the boundaries of everyday human thought. Considering that sentiment, there is a clear connection to the conclusions of “Art and Truth”—the movement beyond a utilitarian or rational interpretation of the Arts. In the chapter’s discussion of the definition of truth in literary analysis, the author raises a point similar to that of “The Truth, The Whole Truth, and Nothing But the Truth” in that our inability to discuss artistic truth is directly linked to our reliance on literal interpretation. Provocative and encouraging a natural empathy, the Arts are placed in a unique position allowing them to address fallacies in human conceptions of themselves and the world.

Introduction to the Arts

Unlike The Arts, Science tells us something valuable about the world.

Science provides an interpretation of the world given in measurable characteristics—the quantitative nature of the sciences allows it a sense of universality not afforded to the arts. This can be gleaned simply from the language used to communicate in the Sciences: there is a standardised form of communicating all observations provided by the International System of Units (SI). How is this valuable? In the increasingly globalised world, modes of communication which extend beyond barriers like culture or location provide a reflection of current states. Through this standardised mode of communication, knowledge of differing origins can cross borders and offer widespread development.

However, the Arts provides a comprehension of culture and qualitative knowledge unavailable to the Sciences. Informing constructions of morals, society, and human nature, artistic knowledge (while more obscure) arguably offers more profound value than the Sciences do in their volatility. The plays of Sophocles have survived over two millennia; the literature of the Tang continues to serve as the basis of cultural studies in East Asian countries. That artistic knowledge has thrived for so long attests to the value that it serves us and the universality of its knowledge—a knowledge that transcends the bounds of time.

Group Verification

“Without the group to verify it, knowledge is not possible.” Discuss.

Group verification is an inherently flawed aspect of knowledge production. When referencing it, we generally refer to the practice of collectively validating a theory or hypothesis. It’s a process reflective of the human history of socialisation, and speaks to the collaborative nature of human knowledge production. Before further discussing the subject, some key questions have been identified, namely the following.

In what ways might group verification be helpful or unhelpful?
Why might we need group verification?

Essentially, this prompt calls for the distinction between personal and shared knowledge. As group verification is an inherently social process, it influences all communicative or social processes of knowledge production. However, one can identify forms of knowledge that do not rely on verification by others—things like instinctual responses or spiritual revelations are deeply individual, but still considered knowledge. For instance, production of knowledge in esoteric religion involves engagement in personal and often uncommunicable experiences of the divine. Without an interloper, a second party, in the process of such knowledge production, it could be considered outside of the group. However, such religions often instead involve small groups of deeply committed individuals striving towards achieving religious enlightenment. A condition such as this highlights the role of community in even throughout individual methods of knowledge production.

Discussing this topic reveals a limitation of the IB TOK framework—would it be possible for personal knowledge to exist beyond the realm of the “Areas of Knowledge” identified by a specific group? Although these areas of knowledge do not claim to encompass all knowledge and do rely on personal processes in the production of new knowledge, they may, in some ways, be constrained by collective definition of their terms. Though their knowledge is not necessarily produced by the group and thus can still exist without it, its validation is impossible. For me, this brings to mind the question of whether or not knowledge needs to be valid or true in order to be considered knowledge. When defined as a “justified true belief”, knowledge takes on alternative characteristics in that it is not a fact or an observation of the surrounding world, but rather validated by the use of reason and logic to support it. This addresses another question that has raised itself throughout my reflections: if a theory or hypothesis is rejected by the masses, but later proved to be true, was it not knowledge when first formulated? (Yet another question: is a theory or hypothesis, in itself, knowledge? While based on pre-existing facts or ideas, can a hypothetical, a possibility, be considered knowledge?) This is digression, however. The key point is as follows.

Community plays such a key role in knowledge production that it is near impossible for a socialised person to produce knowledge without referring to a prior consensus on its validity and accuracy. This itself can be considered a form of group verification, carried out internally in reference to internalised standards of knowledge. Can one consider any socialised individual to truly produce knowledge independently of group verification? On the other hand, does reference to and reliance socialised norms and standards necessarily equate to group verification? One could maintain the stance that, while an individual does rely on their context as a point of origin, the internal nature of personal knowledge prevents it from being wholly subject to group verification.

(to be added: thoughts on the fallacies of knowledge that hasn’t undergone group verification)

Ultimately, we arrive at the conclusion that, while knowledge exists beyond the realm of the interpersonal, group verification is necessary to validate knowledge claims. Oftentimes, the mere classification of something as knowledge is dependent on precedent set by the group. Can knowledge be considered reliable, can its justification be valid, when it lacks the influence of others? Though a point less touched upon within this reflection, it’s important to recognise that group verification is important in ensuring that the conclusions drawn about the world, our justified true beliefs, are well-founded. In the sciences, group verification evaluates methods and encourages scepticism of development to ensure that progression in the field is wholly accurate. Philosophy cannot exist within a vacuum because at the heart of the approach that has defined Western thought, that of the Dialectic method, requires opposing stances to engage in discourse in order to reach the best solution, the synthesis. Without the communal treatment of knowledge, one arrives at an amalgamation of thoughts and impulses governed by little but internal reasoning and the senses.

Competing Hypotheses

The significance of competing hypotheses must be considered when discussing knowledge production in the Natural Sciences. Experimentation in the Natural Sciences ideally comes from combining the inductive and deductive modes of reasoning—the former to produce ideas and hypotheses, and the latter to verify them. In the process of induction, it is possible to identify a variety of hypotheses from the same set of data; it is this quality that has led to the need for a number of different methods to choose hypotheses. One’s choice of hypotheses, as well as one’s rationale, influences the validity of knowledge produced thereafter through the scientific method.

Optimally, a hypothesis should be able to closely explain actual observations, disparate observations, and unexpected data. Considering these three factors, one can generally weigh hypotheses against one another to determine which is the most reliable to depend upon when conducting an experiment. However, in the process of doing so, a number of limitations present themselves. Those which I will primarily focus on are the cognitive biases, which restrict one’s ability to critically consider hypotheses from an objective perspective.

the first approach: occam’s razor;

A common means of determining the best hypothesis is Occam’s Razor, which relies on the Principle of Simplicity in its position that the simplest hypothesis which explains the greatest scope of evidence is most likely to be accurate. In the words of Einstein, “Everything should be made as simple as possible, but not simpler.” As a method, it offers comprehension and prevents misunderstanding by both laymen and fellow experts. The choice of a simpler hypothesis facilitates communication of theories and eases further study. At it’s best, Occam’s razor allows for the elimination of extraneous or invalid hypotheses, and can be considered a guideline for ensuring that the conclusions drawn from testing are relevant and accurate. The reliance on personal preference on the subject of the “simple” can result in unnecessarily complicated or fundamentally illegitimate ideas gaining precedence.

However, a key fault of Occam’s razor lies in its propensity for oversimplification. In the Natural Sciences, a field which aims to describe the world and its processes, its necessary to recognise that the object of study is complicated. The natural world is full of complex processes that cannot be qualified with the simplest explanation. Over-reliance on the law of parsimony could be considered unscientific in that it refutes both logic and intuition in favour of the explanation which best suits individual purpose. Another downfall of Occam’s razor is encouragement of subjectivity—the definition of what a simple explanation is depends on the person establishing a hypothesis. When too many variables are present, as is oft the case in the early stages of establishing a theory, application of Occam’s razor can lead to inappropriate elimination of key ideas, impeding a hypothesis’ efficacy. 

the second approach: multiple hypotheses;

A fitting alternative to Occam’s razor is the use of multiple hypotheses in experimentation, where a range of possibilities are considered when analysing qualitative and quantitative observations. Such a method seeks to facilitate the intuition necessary for the development of the Natural Sciences, encouraging an open-minded and multifaceted approach to scientific discovery. Too often, the reference to a single possibility restricts the bounds of experimentation. When more are considered, they not only increase the productivity of a study, but also encourage the observation of connections between multiple factors and allow for a holistic understanding of the subject of study.

Despite their efficacy in establishing interactions between causes and factors, there are a number of faults that lay in the method of using multiple hypotheses. The practicality of using multiple hypotheses is often limited by resources, making them difficult to test. Without the benefits of unlimited funding, scientists are placed in a position where they must evaluate the most valuable or productive hypotheses to pursue. Beyond a limitation of multiple hypotheses as a method of establishing knowledge, it is a more general factor influencing the production of all scientific knowledge, a process which requires significant time and resources. Another issue with multiple hypotheses is yet another more relevant to the scientists who carry out testing. There is a tendency to allow specific hypotheses to take supremacy when more than one are in consideration. Such a predisposition entirely contradicts the intentions of using multiple hypotheses, and requires scientists to remain aware of their practices and motivations.

conclusions;

Occam’s razor can be considered most efficient in the case of multiple well-established competing theories, by eliminating extraneous details of an idea. However, when it comes to establishing the fundamental elements of a theory, multiple hypotheses allow for a variety of factors to be established and stricken. Using both methods in the appropriate context allows one to arrive at a well-founded conclusion; in fact, it’s quite representative of the marriage between reason and intuition in the Natural Sciences. Through first intuitively establishing a variety of well-founded hypotheses, and then considering them logically using Occam’s razor, we arrive at the most probable and best supported hypothesis (or even hypotheses, if the circumstances permit).