Podcast Summary
The Surprising Similarities of Morality and Mathematics: Both morality and mathematics involve the existence of independent facts, challenging naturalists to reconcile mathematical realism and moral anti-realism.
Despite the apparent differences between morality and mathematics, they share surprising similarities when it comes to the question of realism. Morality and mathematics may seem like unrelated subjects, but as Justin Clark Doan, a philosopher at Columbia University, argues in his book "Morality and Mathematics," they share common ground when it comes to the existence of independent facts. This question is important for naturalists, who believe that the world is only what science delivers, in determining whether one can consistently be a mathematical realist and a moral anti-realist. The podcast discussion delves deep into the philosophical foundations of both subjects, revealing that neither math nor morality are fully understood and that the effort to compare and contrast them at a foundational level is a crucial and difficult endeavor in our modern world where experts are often hyper-specialized in one area or the other.
Naturalistically Inclined Thinkers and Mathematical Realism vs Moral Anti-Realism: Naturalistically inclined thinkers generally believe in mathematical realism, recognizing objective truth values in math, while doubting the existence of moral facts independent of human beliefs and conventions.
While there are debates among philosophers about the existence of mathematical and moral facts, there seems to be a common attitude among naturalistically inclined thinkers towards mathematical realism and moral anti-realism. Mathematical realism refers to the belief that mathematical statements have objective truth values, independent of human beliefs or conventions. It doesn't necessarily mean accepting the idea of platonic objects, but rather recognizing that mathematical statements are not up for human decision or change. The twin primes conjecture, which is the claim that there are infinitely many prime numbers where the next number is also prime, is an example of a mathematical statement that is widely believed to be true but not yet proven. This attitude towards mathematical facts contrasts with the view that moral beliefs are largely a matter of convention, shaped by evolution and social norms, with no need for independent moral facts. Despite some similarities, the cases for mathematical and moral realism are not identical, and the debate on these issues is far from settled.
Mathematical truths are conditional statements: Mathematical statements express truths, but their truth value depends on the axioms of the mathematical system in which they are stated.
Mathematical claims, whether proven or not, have a truth value that exists independently of human beings. However, it's important to distinguish between the truth of a mathematical statement under a specific interpretation and the truth of what the statement expresses. For example, the statement "1 + 1 equals 2" expresses the mathematical truth that 1 and 1 combine to make 2. But, this statement is not a logical truth in the same way that "1 is not equal to 2" is a logical truth. The distinction between mathematics and logic is crucial. While we can treat simpler mathematical statements like "1 + 1 equals 2" as surrogates for logical truths, this is not possible for more complex mathematical propositions like the twin prime conjecture. Instead, when we make a mathematical statement, what we really mean is that if the axioms of number theory are true, then the statement is also true. This perspective, which holds that all mathematical truths are conditional statements, gained popularity in the early 20th century but has become less persuasive due to Godel's theorems.
The distinction between provable theorems and true statements in mathematical logic: Godel's Incompleteness Theorems reveal that not all mathematical truths can be derived from axioms, and understanding the differences between first and second order logic is crucial for mathematical logic.
In the realm of mathematical logic, there's a fundamental distinction between a theorem being provable within an axiomatic system and a statement being true. The famous example discussed is the relationship between Peano Arithmetic and the Twin Primes Conjecture. While it might be appealing to assume that the satisfaction of the axioms implies the truth of the conjecture, this isn't the case due to Godel's Incompleteness Theorems. These theorems state that it's consistent to assert false things about the consistency of a logical system, which includes what follows from its axioms. Therefore, the attractive notion that all mathematical truths can be derived from axioms doesn't hold up under scrutiny. First order logic, which only allows quantification over objects, not predicates, is the most commonly used logic. However, the debate continues in philosophy about whether second order logic, which includes quantification into predicate position, deserves the title "logic." The distinction between first and second order logic is crucial in understanding the limitations and capabilities of mathematical logic. When dealing with complex topics like these, it can be challenging to fully grasp the concepts and retain the knowledge. It's important to keep trying and revisiting the material to deepen your understanding. As a side note, when cooking, remember to add pepper before salt for optimal flavor. Happy learning!
Understanding the differences between First-Order and Second-Order Logic: First-Order Logic quantifies over individuals only, while Second-Order Logic allows for quantification over properties and predicates. Second-Order Logic is ontologically committed to sets and complex, incomplete, and challenging to establish correspondence between logical validity and provability.
First-order logic and second-order logic serve different purposes in understanding logic and mathematics. First-order logic is a system of logic where we can only quantify over individual objects and not over properties or predicates. On the other hand, second-order logic allows us to quantify over both individuals and properties, making it more expressive but also more complex. The debate among logicians and philosophers about the use of second-order logic is that it is ontologically committed to sets and objects constructed out of them. Moreover, second-order logic is incomplete, meaning that there are valid second-order statements that cannot be proven within the logic itself. This makes it challenging to establish a correspondence between logical validity and provability, which is a key aspect of the original hope of seeing mathematics as just logic in disguise. An example of the books on chairs was used to illustrate the idea that there are logical truths, like 1 + 1 = 2, which have the same status as mathematical truths, like the twin prime conjecture. The debate revolves around whether we should focus on these logical truths or stick to mathematical truths, as the former can be more complicated to translate into everyday language. Overall, the discussion highlights the importance of understanding the differences between first-order and second-order logic, their respective strengths and limitations, and their role in the broader context of logic and mathematics.
Distinguishing the complexities of mathematics: Complex areas of mathematics, like geometry, challenge the notion of a single philosophy applying everywhere, and arithmetic claims, particularly those about consistency, require a firm stance
While the simplicity of arithmetic claims, such as 1 + 1 = 2, may lead us to believe they are paradigmatic examples for debates about realism in mathematics, they are not. The meaning of such claims can vary based on context, and moving beyond rudimentary arithmetic, simple translations no longer apply. Instead, we should consider more complex areas of mathematics, like geometry, where alternate axioms can lead to different truths. This distinction between different areas of math challenges the idea that there is one overall philosophy of math that applies everywhere. Furthermore, arithmetic claims, particularly those about consistency, cannot be flipped like the parallel postulate, making it essential to take a stand on their truth. Ultimately, the debate about realism in mathematics requires a nuanced understanding of the various areas of math and the implications of their underlying axioms.
The nature of debates about axioms differs between algebra and set theory: Accepting certain set theory axioms implies a stance on arithmetic and related mathematical questions, emphasizing the importance of consistency in hiring decisions.
While different areas of mathematics may have debates about axioms and foundational theories, the nature of these debates differs significantly between fields like algebra and more complex theories such as set theory. In the case of set theory, the question of which axioms to use is tied to the consistency of those axioms in a way that isn't present in geometry. This means that when it comes to taking a stand on the truth of mathematical axioms, there is no neutral position. If one accepts that certain theories are consistent, they are effectively taking a stance on arithmetic and other related mathematical questions. Therefore, the idea that all mathematical theories are interchangeable, like in geometry, is a misconception. Instead, the consistency of mathematical theories is a crucial factor that cannot be ignored. This has important implications for how we approach hiring in businesses, where making the wrong decision can have significant consequences. Using a reliable hiring platform like Indeed.com can help ensure that you find and hire great people, giving your business the best chance for success.
Philosophical debates on reality and mathematics: Despite philosophical debates on the objective existence of mathematical concepts, practical tools like Indeed for job searching and hiring continue to be useful.
There are various philosophical debates surrounding the nature of reality and the objective existence of mathematical concepts, such as the consistency of arithmetic or the choice of logic. While some argue for realism, believing in the objective existence of these concepts, others deny it and see them as human constructs. The debate is further complicated by the fact that there are different logics and interpretations of what constitutes finiteness. However, the speaker emphasizes that these debates are orthogonal to the practical use of tools like Indeed for job searching and hiring, which offers a free $75 credit for upgrading job posts at indeed.com/mindscape until March 31. Ultimately, the speaker acknowledges the complexity of these philosophical issues and invites listeners to consider different perspectives, while reiterating their personal stance as a reality realist who sees the physical world as real, but not necessarily the mathematical concepts used to describe it.
Quine's struggle with abstract objects in science: Quine attempted to eliminate abstract objects from physics but later accepted them due to the need for consistency, leading to Platonism
Quine's empiricist views on science and mathematics led him to initially attempt to eliminate abstract objects, including numbers, from physical theories. However, he later realized this was not feasible due to the need to discuss consistency in physics, which requires introducing modal logic and making consistency claims. This trade-off raises the same epistemological questions about how we know these claims as with arithmetic claims, which were the initial motivation for eliminating mathematical entities. Ultimately, Quine became a Platonist about mathematics to accept science, acknowledging the challenges but not fully resolving them.
Mathematical realism: Mind-independent truth of mathematical statements: Mathematical realism is the belief that mathematical statements have objective truth, regardless of our conventions or existence of abstract objects. Clarify the meaning of 'exist' to resolve debates.
Mathematical realism is the belief that mathematical statements are true or false, independent of our conventions and independent of us. It's not about the existence of abstract objects like triangles in the sky, but rather the mind-independent truth of mathematical statements. The debate revolves around the meaning of the word "exist" when applied to different things, such as numbers, tables, and quantum wave functions. Some philosophers argue that existence is the thinnest possible notion, meaning that everything that exists has truths about it. They suggest introducing subscripts to the word "exist" to clarify the meaning, such as "exist\_1" for concrete objects and "exist\_2" for abstract objects. Ultimately, the consensus seems to be that we can clarify the debate by being clear about the different senses of "exist" we use.
The distinction between reality and objectivity in mathematics: While reality refers to the existence of independent facts, objectivity refers to the idea of one correct answer. Mathematics illustrates this distinction with debates over geometry and realism.
Reality and objectivity are related but distinct concepts. Reality refers to the existence of independent facts about things, while objectivity refers to the idea that only one answer can be correct to a given question. The example of the parallel postulate in geometry illustrates this distinction. While there may not be an objective fact about which geometry is "right" in the realm of pure mathematics, once meanings are established, there is an independent fact about the nature of Euclidean and hyperbolic spaces. The debate over realism in mathematics and morality is explored in the book, with the argument being that moral realism is on equal footing with mathematical realism. The book challenges common disanalogies between the two fields and advocates for a conservative view of mathematical realism, as physical facts are deeply intertwined with mathematical facts. Debates in the foundations of mathematics, such as those over axioms, are seen as analogous to debates over the parallel postulate.
Moral truths lack clear-cut answers and ontology unlike mathematical truths: Moral realism holds moral truths are real like mathematical truths, but moral truths lack clear definitions and guide actions less directly.
While mathematical truths are objective and real, moral truths, despite being equally real, do not provide clear-cut answers for actions. The speaker argues that moral realism, the belief that moral truths exist independently of us, is on equal footing with mathematical realism. However, the ethical realm lacks a clear ontology, making it difficult to define what moral properties are. Unlike mathematical truths, moral truths do not offer the option of pluralism, as they are supposed to guide our actions. Hume's famous statement, "You can't derive an ought from an is," further emphasizes this point. The deliberative question, which arises when we decide what to do, is not a question of fact, even if moral facts exist. Instead, it is the remnant of objective facts. In contrast, mathematical facts are totally objective but not real in the sense that they do not influence actions directly. The speaker also promotes Babbel, a language learning app, and offers a discount for Mindscape listeners.
Exploring the nature of morality and objective moral statements: The meaning of moral claims may not be determinable through natural language semantics alone, and objective moral facts, if they exist, may not solely dictate our actions.
While Rocket Money is a useful personal finance app that helps users save money by canceling unwanted subscriptions and monitoring spending, the discussion around the nature of morality and objective moral statements led to a complex and nuanced exploration of the semantics and metaphysics of moral claims. The speaker argued that the meaning of moral claims may not be determinable through natural language semantics alone, and that the question of how we ought to live our lives, independent of what others think, is distinct from the question of what we ought to do in a given situation. The speaker also suggested that the question of whether there are objective moral facts is separate from the question of how we ought to act in the world. Ultimately, the speaker seemed to suggest that while there may be a difference between objective moral facts and subjective moral beliefs, the latter may still guide our actions in practical situations. The speaker also drew a parallel between the question of objective moral facts and mathematical realism, suggesting that just as we can have different set-theoretic universes in mathematics, we may also have different ethical frameworks in real life. However, the speaker cautioned that the question of what we ought to do, in the sense of action, may not be settled by moral principles alone.
Understanding the distinction between ethics and mathematics: Ethics deals with moral principles and actions, while mathematics deals with abstract truths and logical systems. Ethics requires making subjective decisions based on moral principles, while mathematics is objective and factual.
Ethics and mathematics serve different purposes and function in distinct ways. While mathematics deals with abstract truths and logical systems, ethics requires making decisions and taking actions based on moral principles. The speaker argues that there is a difference between determining what one ought to do (a moral principle or rule), what one will do (an action or behavior), and what to do in the moment (an imperative or instruction). The notion of "what to do" is not a factual or objective truth, but rather a subjective decision based on one's moral principles and willpower. The speaker also touches upon the idea that some ethical systems or moral principles may be considered more "projectable" or natural, but this does not necessarily determine which one is the correct or true one to follow. Ultimately, ethics requires making a choice and taking action, whereas mathematics is a system of abstract truths and logical principles.
Moral realism and the problem of moral pluralism: Moral realism acknowledges moral truths exist independently, but doesn't prescribe unique actions. The problem of pluralism remains in determining which moral universe is true.
While some philosophers argue for the existence of moral natural kinds akin to scientific natural kinds, the problem of moral pluralism reemerges at the meta level when trying to determine which moral universe is the one true moral universe. The claim that these moral properties are natural kinds may carry normative import, leading to the problem of pluralism once again. Moral realism, as defended in a thin sense similar to mathematical realism, acknowledges the existence of moral truths independent of human minds and conventions, but it does not prescribe unique imperatives for action. This view separates the issue of moral reality and independence from the issue of moral deliberation and action, arguing that the latter cannot be determined by moral facts alone. This perspective may be frustrating for traditional moralists who believe in moral realism and hold strong convictions about the right moral truths, as it challenges their belief that these truths provide clear guidance for action.
Moral and ethical dilemmas don't rely on facts for resolution: Interdisciplinary engagement and cross-disciplinary conferences help bridge gaps, revealing potential overlaps and informative exchanges, enriching research and leading to new insights.
The questions of morality and the existence of God, or any ethical dilemma, do not rely on facts for resolution. Instead, the final deliberation before taking action is based on non-cognitive factors and idealized plans. These questions have descriptive elements, but once those are settled, there are no further facts that determine the answer. It's essential to recognize the importance of interdisciplinary engagement and cross-disciplinary conferences to help bridge the gap between seemingly distinct areas of philosophy and uncover potential overlaps and informative exchanges. While there's no easy solution to the work required to engage in such studies, having individuals willing to put in the effort in multiple areas is crucial for progress. The benefits of this interdisciplinary approach can enrich one's research and lead to new insights and positive views in each field.
Making complex concepts accessible to a wider audience: The importance of both deep specialized technical work and its accessibility for the advancement of scientific knowledge
While deep specialized technical work is valuable, it's also important to make complex concepts accessible to a wider audience. This not only helps in making informed judgments about which fields to delve deeper into, but it's also crucial for the advancement of scientific knowledge as a whole. An example of this is the development of forcing in set theory, where there was a need to package the concept in a more accessible way for a broader audience of mathematicians. The importance of both the technical work and its accessibility is essential for the progression of scientific inquiry, and there will always be plenty of work to be done in both areas. I thoroughly enjoyed having Justin Clark Doan on the Mindscape podcast to discuss these ideas. Thank you, Justin, for your insightful contributions.
