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Logic, Math & Morality

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This is an expansion of a post of mine from another thread. Hat tip to HeKS for bringing the debate around to Ontology vs Epistemology.

In another thread, Seversky rejects the objective nature of morality based on the fact you cannot prove its existence or specific values like you can, say, the speed of light or the gravitational constant. Seversky is making a claim that since there is no satisfying epistemological methodology for establishing – precisely – the good or evil value of a moral proposition, then it must be the case that good and evil, ontologically speaking, are completely subjective commodities.

This is not, of course, a logically valid inference. Just because a commodity has no satisfying or precise, repeatable epistemological methodology for measuring its values doesn’t necessarily mean that the commodity itself is subjective in nature. It just means that we do not have a repeatable, precise means of measuring it. It may be that morality is subjective, it may not be.

The question is not whether we have a repeatable, precise way of measuring the good or evil of a proposition, but whether or not morality represents, in ontological terms, an objectively existent or binding set of commodities (good & evil, or right & wrong).

Seversky said:

There are disagreements which can be resolved by appealing to observations of the natural world and there are those that can’t. The first are about objective issues, the latter are subjective. The debate about whether Newton’s or Einstein’s theories offered a better, more comprehensive account of the physics of the Universe was resolved by observation of that Universe, either directly or by experiment.

Seversky is skipping the part about how such disagreements about the natural world are determined. If logic and math were not considered objectively valid determiners of true statements about the natural world, no such disagreement could ever be resolved. Are mathematics and logic part of the natural world? If so, can anyone direct me to where I can find them? No? Can mathematical or logical principles be objectively proven without using math or logic? No?

How are you going to measure the speed of light without logic or mathematics? Why would you “measure” it at all, or consider it a resolution to the argument about how fast it travels, given that both mathematics and logic themselves cannot be found in the natural world? If they are conceptual and exist only in the subjective minds of individuals, they cannot be accepted as objectively true or universally binding … right?

Is there a satisfying (precise and repeatable) epistemological methodology for determining the validity of logical inferences or mathematical proofs? Not without using the very system one assumes objectively valid in the first place, because math and logic are those things by which such satisfying proofs or evidences are established. We cannot prove math or logic valid because “proving” requires math and/or logic. “Proving” or “demonstrating” requires that we assume the validity of logic and mathematics in the first place. We all agree that at least the fundamental principles of mathematics and logic are objectively binding commodities or (1) we have no means by which to resolve disagreements about the natural world, and (2) we have no means by which to measure the natural world in a way that will resolve disagreements about it.

But, why should we agree that logic and mathematics are, ontologically speaking, objective in nature? They are abstract concepts apparently held in individual minds and do not exist in the physical world in a way we can demonstrate them. If we follow Seversky’s logic, how can we possibly consider them to be objectively binding arbiters when it comes to things in the natural world we physically experience?

As I have argued that we operate as if conscience is a sensory ability that can receive objective moral information, so too are the capacities of logical and mathematical thinking. Even though they are abstract and conceptual, we necessarily act as if those capacities are relaying objective, universally-binding information about reality (whether it is limited to what one defines a “natural” or “physical” world or not). When it comes to logic, math and morality, we necessarily act and must argue as if we are perceiving factual, objective, universally-binding aspects of our existence. We already act, and argue, as if they are, ontologically speaking, objective commodities, not subjective ones. That we cannot prove math, logic and morality reflect objective commodities is irrelevant because sane people necessarily accept them as tools or perceptive capacities which reflect objectively true and binding judgements/measurements.

A disagreement about whether your taste in music is “better” than mine cannot be resolved in that way because taste in music is not a property of objective reality but of our response to that reality. It is subjective.

There is a fundamental difference in how we behave when we are operating under the assumption that our views represent subjective commodities, and under the assumption that our views represent objective commodities. We do not behave as if logic, math and morality are categorically the same as personal preferences; we act as if they represent universally binding, objective measurements/judgements.

(Side note: Seversky seems to think that “our response to that reality” is not itself a part of objective reality. I assume this is just a poor choice of wording from what I assume is a naturalist/materialist/physicalist, although this points to another assumption that is necessary to sane human existence: that we are metaphysically distinct from and transcendent over the matter/energy determinism of the physical world. But that’s another argument for another day.)

Seversky continues:

By the same token, our moral judgements or evaluations of human behaviors in the world are subjective. In physics, the speed of light can be measured to see if it is constant, irrespective of the motion of an observer. There is no similar way to measure whether it is always wrong to kill, regardless of circumstances.

Here, again, is where Seversky confuses an ontological argument for an epistemological one. Nobody has claimed that there is a precise, repeatable epistemological methodology for measuring the values of good or evil. Seversky is making a mistake here thinking that unless a satisfying way exists of measuring a thing (epistemology), it cannot be accepted as objective in nature (ontology). Yet, the very core methodologies Seversky demands be used to demonstrate the ontological nature of a thing (math & logic) cannot themselves be demonstrated objective commodities (ontology). They must simply be assumed objective or else absurdity ensues.

The same is true of morality; if we do not assume that certain core properties of morality are objectively true (love is good, cruelty is evil), then morality is an absurd proposition.

I see no way to escape the subjectivity of morality.

By Seversky’s argument, then, I see no way to escape the subjectivity of mathematics and logic, and so I should disregard them as objective determiners of true or factual statements about the world and start regarding them as subjective preferences. Right?

The way to “escape” the idea that morality is subjective in nature is via correct critical thinking which may require that Seversky, at least for the sake of argument, set aside certain ideological commitments that may be preventing him from a proper rational analysis.

First, he’s thinking about morality incorrectly; it is not proposed as something that can be measured, but rather as something that does the measuring, like logic or math. Morality is like logic or math and conscience is like our capacity to apprehend/perceive logical and mathematical principles and apprehend logical validity and mathematical equations. Yes, we can make erroneous equations and come to logically unsound conclusions, but such disagreements and errors do not mean that logic and math themselves are subjective in nature. There are some things we know are evil just as certainly as we know 1+1=2 or know that A=A. It is undeniable to pain of absurdity to claim that love is evil or cruelty is good. Morality provides sane people with self-evident truths in precisely the same way as do math and logic, and it is from these self-evidently true statements that we must accept, on pain of absurdity, that systems of objectively measuring and judging other things can be and are established.

Second, Seversky needs to recognize that a lack of precise, repeatable epistemological methodology for measuring good and evil (right and wrong) does not mean good and evil are not themselves objective commodities. Just because we cannot precisely-repeatably measure a thing doesn’t mean that we cannot have any credible objective knowledge of a thing at all. We all understand that torturing a child for fun is more evil than not opening a door for an elderly woman (showing her kindness and respect); we all know that unconditional, self-sacrificing and forgiving love is more good than just about any other good one might engage in. Our conscience can perceive general moral measurements we know to be true, even if many moral measurements are difficult to make.

Third, Seversky needs to acknowledge the profound experiential and categorical difference between moral behavior and behavior with regards to subjective preferences and feelings, and that only sociopaths/psychopaths can act as if morality is subjective in nature. He (and others) need to acknowledge that the very idea of forcing subjective feelings/habits/preferences on others is antithetical to the nature of morality. IOW, if morality is nothing more than living according to one’s personal, subjective views and forcing them others when you feel like it, then there is no principled moral difference between Gandhi and Hitler or between the golden rule and might makes right.

“Escaping” the “subjectivity of morality” simply requires being honestly willing to consider the idea that morality represents an objective capacity to judge good and evil, just as one accepts that logic and mathematics represent systems that make objectively valid and universally binding judgements and measurements. One does not have to explain how or why logic, math and morality exist or are objectively binding and valid; one only has to accept that they do exist and are objective in order for their behavior to be rationally consistent with their premises and to avoid existential absurdity.

To deny math, logic or morality represent objectively binding systems of judgement and measurement is to adopt existential, solipsistic anarchism and absurdity. No sane person can even act as if any of those things are subjective in nature. No sane person can even organize a coherent sentence, much less make an argument, that does not reflect the assumption that morality, math and logic are objectively valid, universally binding methods of measurement and judgement.

(However, I doubt Seversky, CF or others can pry themselves away from their ideological commitments sufficiently to honestly consider what it means to deny the objective nature of morality, or to promote the idea that morality is subjective in nature.)

Lacking any evidence to the contrary, even a moral code handed down from on high inscribed on tablets of stone is just another – albeit divine (allegedly) – opinion.

Even though we cannot find any of the principles of mathematics and logic in they physical world, we are capable of recognizing self-evidently true logical and mathematical statements that serve as the axiomatic, objective root of our capacity to make systems of measurement and judgement we consider objective and universal in nature. We cannot prove them; we cannot find them in the physical world; yet it is by their existence that we are able to prove other things and resolve disagreements about other objectively existent commodities.

The same is true of morality. There are self-evidently true moral statements (love is good, cruelty is evil) the denial of which result in absurdity, just like denying that 1+1=2 or denying the principle of identity results in absurdity.

Denying that conscience can perceive objectively valid judgements of good and evil is the equivalent of denying that logic can be used to make objectively valid judgements of truth or denying that mathematics can be used to make objectively valid measurements and equations. It moves one’s worldview into a position that is in absurd contradiction to how they must behave, think and speak, and renders their every moral act the in-principle equivalent to Hitler and Dahmer, by justifying as “good” the forcing of one’s own subjective, personal views on others because, ultimately, one feels like it.

41 Replies to “Logic, Math & Morality

  1. 1
    daveS says:

    WJM,

    Is there a satisfying (precise and repeatable) epistemological methodology for determining the validity of logical inferences or mathematical proofs? No, because math and logic are those things by which such satisfying proofs or evidences are established.

    I think I understand where you’re going here, but perhaps this should be rephrased? There are indeed methods for determining whether mathematical proofs are valid.

  2. 2

    daveS said:

    I think I understand where you’re going here, but perhaps this should be rephrased? There are indeed methods for determining whether mathematical proofs are valid.

    What methods are those, daveS?

    I see what you mean, though, so I edited the above statement in the OP:

    Is there a satisfying (precise and repeatable) epistemological methodology for determining the validity of logical inferences or mathematical proofs? Not without using the very system one assumes objectively valid in the first place, because math and logic are those things by which such satisfying proofs or evidences are established.

  3. 3

    An interesting question here is: why would one assume math and logic objectively binding, and not morality? All three are subjectively held abstract concepts; all three rely upon non-demonstrable, so-called “intuitions” about correct thinking, all three demand that we act, argue and think as if they are indeed objective commodities.

    So, why is it any easier to accept logic and mathematics as objectively binding systems of judgement and measurement, and not morality, when we can state moral principles that are as self-evident and as indispensable to our experience and behavior as 1+1=2 or A=A?

  4. 4
    kairosfocus says:

    WJM, attn Sev:

    [Sev:] There are disagreements which can be resolved by appealing to observations of the natural world and there are those that can’t. The first are about objective issues, the latter are subjective.

    This fails, most likely by equating empirical experience of the physical world with reality.

    WJM is right to point to mathematics as an abstract, objective domain of knowledge in answer.

    Math is the study of the logic of structure and quantity, which is wholly abstract, is accessed by subjects using their faculties for rational, responsible thought and is objective. That is, the findings and grounding warrant transcend the particular perceptions, views, likes, tastes and experience of any one individual or group.

    And at the core of grasping the reality of morality and Mathematics alike there are self-evidently true assertions that are seen as true on pain of absurdity. Nor are they unconnected. In fact the sense of urgency towards mathematical truth is just as much an expression of the voice of conscience as the sense of urgency towards truth in general.

    There is deep wisdom in the old saying that all [genuine] truth is God’s truth.

    KF

    PS: There are also objective principles of aesthetics. Indeed, with the Pythagoreans, it was discovering the mathematical structure of music that led them to look for explanation of the harmony of the heavens in the same domain, and onward to the mathematical vision of the sciences. So, no, it is not true that beauty is a chaotic, irrational wholly subjective phenomenon.

  5. 5
    kairosfocus says:

    PPS: should I mention the simple but useful RION framework for scales of measurement, and that comparative judgement is involved directly or indirectly in measuring? Ratio, interval, ordinal, nominal? The presence of Likert scales or polytomous Rasch scales for ordering that routinely appear in contexts of evaluation?

  6. 6
    ellazimm says:

    WJM

    Is there a satisfying (precise and repeatable) epistemological methodology for determining the validity of logical inferences or mathematical proofs? Not without using the very system one assumes objectively valid in the first place, because math and logic are those things by which such satisfying proofs or evidences are established.

    I’ve got this feeling you haven’t seen many mathematical proofs. But if I’m wrong I’d love to hear about your experience.

  7. 7
    Querius says:

    WJM wrote:

    It moves one’s worldview into a position that is in absurd contradiction to how they must behave, think and speak, and renders their every moral act the in-principle equivalent to Hitler and Dahmer, by justifying as “good” the forcing of one’s own subjective, personal views on others because, ultimately, one feels like it.

    Relativism was once used to attack the absolute Judeo-Christian ethics. Once victorious, relativism adopted a set of supposedly self-evident, immutable truths that are ferociously enforced by political correctness mobs on campus. No dissenting rational discussion or debate is tolerated.”She’s a witch” has been replaced with “he’s a fascist.” The irony is completely lost on them.

    However, the way to defend yourself against one these self-appointed vigilantes of political correctness is easy, but usually overlooked.

    As you’ve seen here on occasion, appeals to logic, pointing out contradictions, or presenting historical facts are all completely ineffective.

    What is highly effective is to take up the Shield of Genetic Fallacy to deflect any attacks and the Sword of PC Orthodoxy to find in them the smallest deviation in PC conformity, and to start hacking them on that point alone without mercy until it becomes apparent to their friends that they are heretics. Their former friends will then eat them alive.

    No one escapes the new Inquisition!

    -Q

  8. 8
  9. 9
    Jon Garvey says:

    Always like this quote from Sir Arthur Eddington, who of course was smarter-than-the-average-bear on both mathematics and the speed of light, having been the one to confirm Einstein’s theory by observation:

    “Dismiss the idea that natural law may swallow up religion; it cannot even tackle the multiplication table single-handed.” Swarthmore Lecture in 1929.

    He refers, of course, to the metaphysical assumptions that must, necessarily, underlie maths, and even more empirical science.

  10. 10

    ellazimm said:

    I’ve got this feeling you haven’t seen many mathematical proofs. But if I’m wrong I’d love to hear about your experience.

    I’ve got this feeling you don’t have anything of substance to add to this thread. But if I’m wrong I’d love to hear your salient rebuttal to any point in the O.P.

  11. 11
    Andre says:

    I wonder

    Has Ellazimm ever heard of Gödel’s incompleteness theorems?

  12. 12
    bornagain77 says:

    WJM, I think you may appreciate the topic of this post from crev.info since it runs along the sames lines of refutation that you are so apt at using in these debates with atheists:

    Imploding Ideas Unnoticed by Their Advocates – June 5, 2016
    http://crev.info/2016/06/imploding-ideas/

    One thing in particular caught my eye from that article:

    Explanation implies importing truth. If truth evolves, it isn’t true any more. And explaining “in a better way” presupposes an unevolving standard: i.e., knowing which explanation is better and which is worse—not just for our culture, but for all time.

    After reading this, it stuck me that, since atheists hold that randomness is the creator of all things, even their own illusory thoughts, then this prevents them from ever arriving at that ultimate unchanging ‘truth’. Crev.info puts it like this:

    He speaks as a guru whose eyes, like the Buddha, have been enlightened. A clever debater could ask if his thinking has stopped evolving yet and reached Nirvana, and how would he know. If it is still evolving, who’s to say it might evolve into its opposite some day?

    WJM, I’m pretty sure you can see the devastating contradiction in the foundation of the atheistic worldview as to ever attaining the real truth. Hopefully in the future, in your eloquent manner, you can hash this devastating contradiction in the very foundation of the naturalist’s worldview out more clearly.

    A few related quotes:

    Do the New Atheists Own the Market on Reason? – On the terms of the New Atheists, the very concept of rationality becomes nonsensical – By R. Scott Smith, May 03, 2012
    Excerpt: If atheistic evolution by NS were true, we’d be in a beginningless series of interpretations, without any knowledge. Yet, we do know many things. So, naturalism & atheistic evolution by NS are false — non-physical essences exist. But, what’s their best explanation? Being non-physical, it can’t be evolution by NS. Plus, we use our experiences, form concepts and beliefs, and even modify or reject them. Yet, if we’re just physical beings, how could we interact with and use these non-physical things? Perhaps we have non-physical souls too. In all, it seems likely the best explanation for these non-physical things is that there exists a Creator after all.
    http://www.patheos.com/Evangel.....#038;max=1

    “One absolutely central inconsistency ruins [the popular scientific philosophy]. The whole picture professes to depend on inferences from observed facts. Unless inference is valid, the whole picture disappears… unless Reason is an absolute, all is in ruins. Yet those who ask me to believe this world picture also ask me to believe that Reason is simply the unforeseen and unintended by-product of mindless matter at one stage of its endless and aimless becoming. Here is flat contradiction. They ask me at the same moment to accept a conclusion and to discredit the only testimony on which that conclusion can be based.”
    —C.S. Lewis, Is Theology Poetry (aka the Argument from Reason)

    Verses and video:

    Malachi 3:6
    “For I, the LORD, do not change; therefore you, O sons of Jacob, are not consumed.

    Hebrews 13:8
    Jesus Christ is the same yesterday and today and forever.

    John 14:6
    Jesus answered, “I am the way and the truth and the life. No one comes to the Father except through me.

    (Centrality Concerns) The Resurrection of Jesus Christ from Death as the “Theory of Everything” – video
    https://www.facebook.com/philip.cunningham.73/videos/vb.100000088262100/1143437869002478/?type=2&theater

  13. 13

    I appreciate the kind words, BA77. You have the best resources! Always appreciate your links & input. I’ll check it out ASAP.

    Many materialists/physicalists/subjectivists/atheists have long since moved beyond the reach of reason (as recent threads have demonstrated), but perhaps these and future posts can help some who haven’t gone that far yet.

  14. 14
    daveS says:

    WJM,

    There is in fact quite a bit of subjectivity in mathematics and logic (see here for some examples). FWIW, I don’t think of mathematics as a body of objective truths, and I think many mathematicians would agree.

  15. 15
    Barry Arrington says:

    daveS, is it objectively true that if one has a set with a cardinality of two and a separate and distinct set with a cardinality of two and then one combines the two sets that the resulting set has a cardinality of four?

    Let me help you with that. The answer is yes.

    WJM pointed to math as an abstract enterprise that contains objective truths that may be denied only on pain of absurdity. I just demonstrated one such. Your effort to deflect, dissemble and deny fails miserably. In fact, you should be ashamed of the attempt. I doubt that you are.

  16. 16
    daveS says:

    Barry,

    Note that I didn’t say that there aren’t any objective truths in mathematics or logic. In fact, I’m not even arguing there are no objective moral truths.

    No deflection, dissembling, or denial intended.

  17. 17
    john_a_designer says:

    Another question that can be raised is whether or not the naturalist has grounds for any kind reliable knowledge. In other words, if we cannot have any kind of real (objective) moral truth can we have any kind of truth at all? This takes us to the heart of epistemology itself. In my view the subjectivist has no ground in cherry picking moral knowledge as something that cannot be grounded in truth without casting doubt on any kind of truth claim— even, if not especially, his own philosophical truth claims.

    Modern naturalists believe that the human mind with its knowledge and beliefs is the result of a mindless evolutionary process. How can a mindless process give us reliable knowledge and beliefs?

    Charles Darwin appears to have also been disturbed by this question. He wrote in a letter to a friend:

    “With me,” says Darwin, “the horrid doubt always arises whether the convictions of man’s mind, which has been developed from the mind of the lower animals, are of any value or at all trustworthy. Would anyone trust in the convictions of a monkey’s mind, if there are any convictions in such a mind?”

    Patricia Churchland, a philosopher who specializes in issues raised by cognitive science, has argued that the way our nervous system and brain evolved they cannot be expected to give reliable knowledge and beliefs.

    “Boiled down to essentials, a nervous system enables the organism to succeed in the four F’s: feeding, fleeing, fighting, and reproducing. The principle chore of nervous systems is to get the body parts where they should be in order that the organism may survive. . . . . Improvements in sensorimotor control confer an evolutionary advantage: a fancier style of representing is advantageous so long as it is geared to the organism’s way of life and enhances the organism’s chances of survival [Churchland’s emphasis]. Truth, whatever that is, definitely takes the hindmost.”

    According to retired University of Notre Dame philosopher Alvin Plantinga, “Darwin and Churchland seem to believe that (naturalistic) evolution gives one a reason to doubt that human cognitive faculties are reliable (produce mostly true beliefs): call this ‘Darwin’s Doubt’.”

    Plantinga, on the other hand, argues that:

    “The traditional theist… has no corresponding reason for doubting that it is a purpose of our cognitive systems to produce true beliefs, nor any reason for thinking the probability of a belief’s being true, given that it is a product of her cognitive faculties, is low or inscrutable. She may indeed endorse some form of evolution; but if she does, it will be a form of evolution guided and orchestrated by God. And qua traditional theist — qua Jewish, Moslem, or Christian theist – she believes that God is the premier knower and has created us human beings in his image, an important part of which involves his giving them what is needed to have knowledge, just as he does.”

    In other words, theism provides a sufficient foundation for truth and knowledge, naturalism/materialism does not.

  18. 18

    daveS said:

    There is in fact quite a bit of subjectivity in mathematics and logic (see here for some examples). FWIW, I don’t think of mathematics as a body of objective truths, and I think many mathematicians would agree.

    Then said:

    Note that I didn’t say that there aren’t any objective truths in mathematics or logic. In fact, I’m not even arguing there are no objective moral truths.

    I don’t understand your contribution, daveS. Are you contesting some point I made? If so, what point are you contesting (please quote it) and tell me explicitly what your challenge to it is.

  19. 19
    daveS says:

    WJM,

    I don’t understand your contribution, daveS. Are you contesting some point I made? If so, what point are you contesting (please quote it) and tell me explicitly what your challenge to it is.

    Looking back at your post, I might have misinterpreted this, which is what I was responding to:

    No sane person can even organize a coherent sentence, much less make an argument, that does not reflect the assumption that morality, math and logic are objectively valid, universally binding methods of measurement and judgement.

    Do you accept that logic and mathematics do have subjective components? For example, the validity of a logical or mathematical proof depends on the axiomatic system one is working in. And it’s not clear to me that whether certain axioms are actually true can be decided objectively.

    If that is your view, I don’t disagree.

  20. 20
    ellazimm says:

    WJM

    I’ve got this feeling you don’t have anything of substance to add to this thread. But if I’m wrong I’d love to hear your salient rebuttal to any point in the O.P.

    I just want to know how much experience you’ve had with mathematical proofs. Why is that so hard to address?

    Andre

    Has Ellazimm ever heard of Gödel’s incompleteness theorems?

    Yup, I have. Why do you think it’s pertinent to this discussion?

    From Wikipedia:

    The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an “effective procedure” (i.e., an algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.

    I like this briefer version:

    Hilary Putnam (1960) suggested that while Gödel’s theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. Assuming that it is consistent, either its consistency cannot be proved or it cannot be represented by a Turing machine.

    Note the caveat: assuming consistency. And note that, given a consistent system, Godel is just saying you can’t prove it’s consistent or you can’t represent it as a Turning machine.

    What you can do, as with non-Euclidean geometry, is vary the axioms.

  21. 21

    Ellazimm said:

    I just want to know how much experience you’ve had with mathematical proofs. Why is that so hard to address?

    How much experience I’ve had is irrelevant. Let’s assume I have none. What is germane to the argument is whether or not what I have argued is valid, not whether I have any experience in what it is I am arguing about. A valid argument made by a fool is still a valid argument nonetheless.

    Now, if you have a rebuttal or counter-argument to make about anything I’ve actually said or argued in this thread, feel free to present it and demonstrate my ignorant foolishness. Otherwise, stop trolling my thread.

    daveS said:

    Do you accept that logic and mathematics do have subjective components? For example, the validity of a logical or mathematical proof depends on the axiomatic system one is working in.

    daveS, how would you determine the subjectively proposed axioms of a mathematical and/or logical structure – meaning, what determining system would you use? Then, once you’ve established what the axioms and inferences are within that subjectively-constructed “if-then” scenario, how would you establish whether or not that system had internal validity? How would you establish whether or not a mathematical equation that uses subjectively supplied variables or subjectively chosen starting points utilizing some subjectively chosen frame of reference had reached an internally correct solution?

    Does one apply principles formulas and laws axiomatically assumed to be objectively valid and binding to check such work for errors or mistakes, and to verify internal validity? Or can we just make up our own idiosyncratic, subjective system of mathematical and logical rules and insist that others use it when checking our work?

    Do you have an actual argument to make here daveS, or are you just trolling for some phrase you can latch on to have a problem with?

    And it’s not clear to me that whether certain axioms are actually true can be decided objectively.

    How would one “decide objectively”, daveS? What system of evaluation would they use to make such a determination? Did you not get the part in the OP where I said:

    We cannot prove math or logic valid because “proving” requires math and/or logic. “Proving” or “demonstrating” requires that we assume the validity of logic and mathematics in the first place. We all agree that at least the fundamental principles of mathematics and logic are objectively binding commodities or (1) we have no means by which to resolve disagreements about the natural world, and (2) we have no means by which to measure the natural world in a way that will resolve disagreements about it.

    Do you have a bone to pick with that, daveS? If so, please make your objection explicit. Please refrain from making vague, seemingly quarrelsome remarks; it reads like trolling.

  22. 22
    daveS says:

    WJM,

    I’m honestly not sure what you think I’m arguing about.

    I will answer your questions in detail if you like, but first, are you affirming that you do believe there are subjective components in mathematics, and that you essentially said that in the OP? If so, then as I said above, I agree with you on that.

    Here’s an example: The Axiom of Choice. Some believe it is true, some say it’s false, some say it’s meaningless (and probably some don’t care). That’s a philosophical matter which must be addressed, but as far as I know, there is no “objectively correct” answer. Some mathematicians accept AoC and use it, some reject it.

    If you believe this is an example of subjectivity in mathematics, I have no quarrel with you on this matter, as I said in #19.

  23. 23
    Querius says:

    Heh, I like the way that Ellazimm assigns homework to everyone in 20.

    IMO, the problem with mathematics in general is applicability, boundary conditions, and the limits to extrapolation. Your math might be fine, but applying it wisely requires some insight. Thus, mathematics can be congruent to truth, but not of necessity. This is related to Gödel’s theorems in a sort of inverse way.

    But we’re looking for objective truth, and math seems to be a likely place since there’s not a lot of controversy about sums, for example. But then, what’s the popular proverb about statistics?

    Or maybe it’s not objective truth that we’re looking for, but for wisdom and objective morality. Jesus had some pithy things to say about about the subject.

    Also, as I used to tell our children, in ancient times, television had a different name—it was your full-color, high-definition, wide-screen “window” overlooking the main street. Reality shows comprised watching other peoples’ lives either unfold or unravel.

    Is your life unfolding or unraveling? Something to think about.

    -Q

  24. 24
    ellazimm says:

    WJM

    How much experience I’ve had is irrelevant. Let’s assume I have none. What is germane to the argument is whether or not what I have argued is valid, not whether I have any experience in what it is I am arguing about. A valid argument made by a fool is still a valid argument nonetheless.

    I think your argument shows your inexperience, a lack of understanding. I think you don’t really grasp the systems you are discussing. You don’t build on work that has already been done. So I don’t take you seriously. It would be like me making a theological argument; I would quickly show my lack of ability.

    Querius

    But we’re looking for objective truth, and math seems to be a likely place since there’s not a lot of controversy about sums, for example. But then, what’s the popular proverb about statistics?

    Mathematics is an axiomatic system. ‘Objective’ truth is not its goal. Some parts do have (sometimes surprising) real-world applications. Many parts have absolutely noting to do with reality. Some people, like KairosFocus, are uncomfortable with certain aspects of mathematics because they feel they run counter to their theology. That’s not the way it works.

    First you do the mathematics. Then you decide if it has anything to do with physical reality. If your goal is to model a situation then you start from a different point. Then you’re trying to find something that is applicable. But the structure you find that makes the model work came about, usually, without that model in mind.

    It’s not like it was 2000 years ago when real-world applications were the primary ‘research’ focus. Everything’s different now. You don’t talk about ‘objective’ truth in mathematics.

  25. 25
    ellazimm says:

    WJM

    how would you determine the subjectively proposed axioms of a mathematical and/or logical structure – meaning, what determining system would you use? Then, once you’ve established what the axioms and inferences are within that subjectively-constructed “if-then” scenario, how would you establish whether or not that system had internal validity? How would you establish whether or not a mathematical equation that uses subjectively supplied variables or subjectively chosen starting points utilizing some subjectively chosen frame of reference had reached an internally correct solution?

    Like I said, your lack of understanding shows. The questions you ask prove it.

  26. 26
    Querius says:

    Ellazimm @24 wrote:

    First you do the mathematics. Then you decide if it has anything to do with physical reality. If your goal is to model a situation then you start from a different point. Then you’re trying to find something that is applicable. But the structure you find that makes the model work came about, usually, without that model in mind.

    And that was exactly the case with regard to the course in non-euclidean geometry that I once took. (I love it when people help me make my point).

    Why should mathematics be any different than systems of logic? Existential logic is also axiomatic. From Gödel’s theorems, we know that there’s no single mathematical system that will lead to all possible true statements. And you agreed that mathematics may lead to statements that cannot be applied to physical reality (“truth”). This is why I don’t care for Spinoza’s attempt to create a system of ethics based on first principles.

    Ok, so where does that leave you?

    Creating or adopting an arbitrary ethical system will certainly conflict with those created by others.

    Who will arbitrate?

    For example, on what grounds were the Nuremberg trials convened. “We have defeated you by force of arms, so therefore we have the right and the imperative to put you on trial followed by execution or imprisonment.”

    You ask for a commonly understood criteria of morality and ethics, but I won’t agree to that. How do you determine the moral difference between frying and eating the embryo of a chicken to do the same with another animal species—human, for example?

    -Q

  27. 27

    ellazimm said:

    I think your argument shows your inexperience, a lack of understanding. I think you don’t really grasp the systems you are discussing. You don’t build on work that has already been done. So I don’t take you seriously. It would be like me making a theological argument; I would quickly show my lack of abilit

    Like I said, your lack of understanding shows. The questions you ask prove it.

    Until you actually rebut anything I’ve argued, EZ, all you are doing here is trolling. Either demonstrate the flaws in my argument or stop trolling my thread. All you are doing here is avoiding any actual engagement of my argument.

    You see, I don’t think you have any comprehension whatsoever about the actual argument I’m making, EZ. I think you just found a term that relates to something you have some expertise in that you can troll in order to score rhetorical points without ever actually addressing the argument.

    Either respond to the argument or stop trolling. I won’t warn you again.

  28. 28
    john_a_designer says:

    What motivates trolls? Fear? Contempt? Smug condescension? A combination of the three? Something else? My motive is to seek the truth. For me, unlike many Christian-theists, faith does not trump truth; truth trumps faith. In other words, if you could give me a strong enough reason backed by objectively true evidence, I would disavow my beliefs. Whatever I believe I want it to be really true.

    Why are trolls afraid to take the same approach? Why do the waste so much time (theirs and ours) on pretension and posturing? Do they really expect people to change their beliefs because of their smug condescension? If nothing else that proves to me that they have nothing of substance to offer.

  29. 29
    kairosfocus says:

    EZ,

    BTW I find I must correct what is in fact a strawman caricature. (And there is a similar suggestion in the DS reference somewhere to “subjectivity” on views of Math, with a link to a thread where the following was discussed.)

    The ascription of a theological motive for my discomfort with how we have handled infinity is so far wrong that I am astonished.

    My concern is that the commonly presented result that here is an endless — infinite — succession of finite +1 increment counting numbers from 0, 1, 2 on that is spanned by a procedure that points across an ellipsis of endlessness, is LOGICAL.

    Infinitely many successively attained FINITE successive + 1 numbers from 0, 1, 2 on through a chaining process is to my mind now clearly a plain old fashioned contradiction. At first I wondered, but now I am pretty sure of it.

    That is why I took time to construct a model, based on computer punch tapes, with parallel pink and blue tapes that are endless to the right. Advance the blue one by any large value K, that is finite, and we see that we may endlessly match 0, 1, 2 . . . with k, K+1, k+2 etc with the same endlessness tot he right. As 0, 1, 2 . . . is endless and infinite, k, k+1, k+2 etc is in 1:1 correspondence and is also infinite, of countable cardinality.

    The conclusion I drew is that no finite increment successive procedure can span and complete an endless range, we will at any finite stage k still have a countable infinity onwards.

    Therefore, ENDLESSNESS must be taken seriously, and understood as not span-able in finite increment successive steps.

    This then implies for ordinary induction that what it proves is that for any value of succession we can attain, the relevant chaining property will hold once it holds for an initial value also. But this does not span the endlessness in steps.

    Transfinite induction is required to do that.

    beyond, I point out that we can see too that if we impose a decrementing process on one side of an ellipsis of endlessness with . . . 2, 1, 0 at the other end, the same obtains, as we can do a similar exercise with decrementing. (Surreal numbers are an obvious way to set that up.)

    That does have a significant metaphysical import, that one cannot descend to now by a postulating an infinite past chain of past causal succession of definite discrete finite duration stages. (No, Zeno type paradoxes on the continuum are irrelevant.) But the import is rooted in the logic of quantity and structure, not in a metaphysical a priori arbitrarily imposed on things that otherwise are unproblematic.

    The metaphysical import comes up in connexion with a linked issue that atheistical metaphysics has to either postulate an infinite causally successive past or else that at some point in the past we got a cosmos out of a genuine nothing, non-being.

    neither of these is attractive, but it seems that infinite descent is seemingly more plausible.

    It fails.

    w+a, w +(a-1), w+ (a-2) . . . w . . . 2, 1, 0

    faces that w+a, w +(a-1), w+ (a-2) . . .

    can readily be matched with 0, 1, 2 . . . and with the ellipsis of endlessness intervening, we run into the unbridgeable span.

    We can point across it logically and just as we assign an order type successor, we can assign an order type immediate or remote as we wish, predecessor.

    But — as language is trying to tell us — we cannot span endlessness in finite stage cumulative, successive steps. So spanning and ending the endless is a fallacy of self-contradiction.

    We have to take the ellipsis of endlessness seriously.

    I hope that if I am being misrepresented elsewhere as you just summarised, this will now be corrected.

    Just a position statement, given a blatant misrepresentation.

    This is my view, with given reasons.

    If I am wrong, it should be readily shown how such a transfinite span of endlessness can be spanned in successive, cumulative, finite stage steps, e.g. +1 steps, and how there is an infinite number of finite numbers successive from 0, 1, 2, given that each successor k is in effect the order type of the set so far from 0 to k-1, i.e. there is a copy the set so far principle: { } –> 0, {0} –> 1, {0,1} –> 2 etc. To my mind, that means that for an endless succession, the set so far at some stage would have to reach to endlessness, while having finite order type.

    That seems contradictory, a better conclusion is we can only ACTUALLY succeed to finite k, but every k we reach is succeeded by onward endlessness that may be matched 1:1 with the set from 0:

    0, 1, 2, . . .

    k, k+1, k+2, . . .

    So, we see that every counting number we can attain to is finite and bounded, but the set goes on endlessly beyond such a point k.

    As there is no completion in steps, the ellipsis of endlessness is key to understanding the set of counting numbers and we are not warranted to conclude that we have an endless, completed succession of FINITE numbers.

    If you want to take up this debate again, there is an already existing thread where there was a very long exchange.

    Though, I doubt that this will go much beyond this point.

    I trust this is not too far removed from the logic and math part of the focus of this thread at least.

    KF

  30. 30
    Barry Arrington says:

    ellazimm @ 24. You really are quite shameless, as WJM has already demonstrated at 27.

    I am of two minds about whether to ban you.

    I am tempted to do so, because your bad faith attempts to deflect and dissemble are really annoying to the grownups trying to discuss these matters.

    OTOH, it is always handy to have a foil, someone from the other side who comes in and makes really stupid comments that can then be contrasted with the logical and reasonable arguments from the ID side.

    For now, I am leaning toward “foil.” Ain’t it good to know that you serve a purpose, even if that purpose is “bad example of how to argue.”

  31. 31
    daveS says:

    KF,

    BTW I find I must correct what is in fact a strawman caricature. (And there is a similar suggestion in the DS reference somewhere to “subjectivity” on views of Math, with a link to a thread where the following was discussed.)

    If you’re referring to a strawman caricature somewhere in my post, I don’t believe such exists. I was just pointing out that difficult philosophical issues arise in mathematics, and as far as I can tell, there is no objective way to resolve them. For another example, take the Continuum Hypothesis. Is it true, false, neither? Whether you choose to accept it as an axiom seems to me to be a highly subjective matter.

  32. 32
    kairosfocus says:

    DS, was it your reference that used a link to the thread spoken of? Which does not discuss the continuum hyp but what was just outlined from my side? On CH, my thought is, the objectivity is, that so far there is no basis for the choice being forced to one particular side (as in independent of ZFC), so there is a bifurcation and it is worth going down both sides. KF

  33. 33
    daveS says:

    KF,

    DS, was it your reference that used a link to the thread spoken of? Which does not discuss the continuum hyp but what was just outlined from my side?

    Yes. In that thread all sorts of positions and issues came up, such as predicativism, finitism, intuitionism, and so on. I’m not implying anything negative (or positive) about any of these positions, just that they exist.

    To be sure there was no in-depth discussion of CH in that thread; it’s just one more example I thought of.

    On CH, my thought is, the objectivity is, that so far there is no basis for the choice being forced to one particular side (as in independent of ZFC), so there is a bifurcation and it is worth going down both sides. KF

    I take the same position. However, a platonist would (might?) believe that it’s either true or false, so the correct answer is out there waiting to be discovered, while a nominalist would disagree.

  34. 34
  35. 35
    daveS says:

    Indeed.

  36. 36
    clown fish says:

    Barry: “OTOH, it is always handy to have a foil, someone from the other side who comes in and makes really stupid comments that can then be contrasted with the logical and reasonable arguments from the ID side.”

    I thought that was my job. Barry, you aren’t looking to replace me, are you? 🙂

  37. 37
    ellazimm says:

    [Do not troll my thread, EZ. You were warned twice.]

  38. 38
    kairosfocus says:

    EZ, I have specifically informed you about the nature of my actual discomfort and opinion; which is logical and conceptual. KF

    PS: Likewise, you will notice that when I have headlined a topic for record, it is connected to an active discussion thread so there is opportunity for discussion.

  39. 39
    john_a_designer says:

    Here are some interesting comments by mathematician Roger Penrose from a Closer To Truth interview which was conducted recently by Robert Kuhn. The excerpts are from a transcript of a podcast that William Lane Craig did about the Penrose’s interview. I think Penrose says a number of things about the real truth and real world applicability of mathematics that are relevant here.

    I think often people take the view that there is another kind of reality which is the mental reality. Certainly philosophers might have that view. Some might even regard the mental world as being in some sense primary and the physical world is somehow to be thought of as a construct from mentality. I don’t particularly like that view. In my view you have to think of a third one. I am sometimes accused of being not just a dualist but actually a trialist, which is even worse…

    [M]athematics seems to have its own kind of existence. It is very important in understanding the physical world that our way of describing the physical world, certainly at its most precise, has to do with mathematics. There is no getting away from it. That mathematics has to have been there since the beginning of time. It has eternal existence. Timelessness really. It doesn’t have any location in space. It doesn’t have any location in time. Some people would take it not having a location with not having any existence at all. But it is hard to talk about science really without giving mathematics some kind of reality because that is how you describe your theories in terms of mathematical structures…

    It also has this relationship to mentality because we certainly have access to mathematical truths. I think it is useful to think of the world as not being a creation of our minds because if we do then how could it have been there before we were around? If the world is obeying mathematical laws with extraordinary precision since the beginning of time, well, there were no human beings and no conscious beings of any kind around then. So how can mathematics have been the creation of minds and still been there controlling the universe?

    (emphasis added by me)

    Read more: http://www.reasonablefaith.org.....z4Ax98Y4cs

    Here is another relevant quote from the part two podcast.

    There is another feature about this which is that in each case it is only a small part of one world which encompasses seemingly the entirety of the next world. Only a little bit of mathematics – a very subtle, beauty, and powerful part of mathematics, but there is the whole world of mathematics . . . if you look at any pure mathematical journal it is full of things that have absolutely no relevance to physical activity. OK maybe the odd thing turns out to have, but most of it seems to have no relation to physical reality. That is not the point of it. You do it because you are interested in mathematics for its own sake. So it is a small part of the mathematical world which seems to encompass the behavior of the physical world. And it is a very small part of the physical world which seems to evoke mentality. There are far more rocks and things – dead planets – around then there are conscious brains around. It is only a very small part of that. It has to be organized in a very subtle and sophisticated way to give rise to mentality whatever that is. And it is only a small part of our mental activities which relate to mathematics. I think most non-mathematicians would appreciate that. But even mathematicians most of the time are thinking about other things. So it is only a small part of our mentality. I like to draw this picture almost in a paradoxical way. Each world is a small part of it which seems to encompass the next one as you go around.

    Read more: http://www.reasonablefaith.org.....z4AxJgIaP4

    It appears to me Penrose, who is BTW not a theist, is pretty much agreeing with WMJ about the real mind independent truth (objectivity) of mathematics. Of course, I am aware that his is not the only meta-mathematical view held by leading mathematicians.

    But whose opinion should I put more weight on? Dr. Penrose who takes the time to explain his position or the anonymous trolls and drive-bys, who show up here and proclaims smugly to WMJ,

    “I think your argument shows your inexperience, a lack of understanding. I think you don’t really grasp the systems you are discussing. You don’t build on work that has already been done. So I don’t take you seriously.”

    He is not building “on work that has already been done?” Says who?

  40. 40
    daveS says:

    john_a_designer,

    The part where Penrose describes a third mathematical “world” is quite interesting. I will have to read more about that.

  41. 41
    john_a_designer says:

    Oops, it appears that I got William J Murray’s initials wrong. Obviously it is WJM not WMJ. However there might be “Freudian slip type reason” for my mistake. WMJI (“magic”) were the first three call letters of an oldies radio station that I used to listen to. Force of habit? Or, it was just a dumb goof. Who knows? I don’t even know. As Hamlet said to Horatio, “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.” So true, so true…

    However, it is kind of scary when you don’t even know how your own mind works.

    One thing that I should have emphasized. WJM wrote in his OP:

    ”But, why should we agree that logic and mathematics are, ontologically speaking, objective in nature? They are abstract concepts apparently held in individual minds and do not exist in the physical world in a way we can demonstrate them.

    In other words, they are mysteries. Penrose agrees. He writes that “there is the relationship between these three worlds[the physical world, mental world and mathematical world] which I regard… as somewhat mysterious or very mysterious. I sometimes refer to this as ‘three worlds and three mysteries.’ Mystery number one is how is it that the physical world does in fact accord with mathematics, and not just any mathematics but very sophisticated, subtle mathematics to such a fantastic degree of precision…”

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