Is Math Discovered Or Invented?

Mathematics is one of those things that deeply fascinates me but that has always gone over my head no matter how much I try to study it. Among its most intriguing qualities is whether it has any independent existence outside of human observation or thought, or whether it is wholly or largely an artificial concept.

The following five minute video briefly covers this intriguing question. I found it fairly easy to digest despite my ignorance of the subject.

If anyone out there is well versed in mathematics, feel free to weigh in with your knowledge?

Can You Change Your Mind?

We all have beliefs and opinions that will likely remain unchanged no matter what counter-evidence is brought to our attention. This is actually typical of all humans, since our politics, faith, values, and views are shaped by psychological and social conditions that are largely outside of our control (and usually unnoticed in their influence).

Yet every one of us will claim that our beliefs are based on sound reasoning and facts — in contrast to our opponents, of course. In that case, we should ask ourselves the following: what would it take for me to accept my opponent’s beliefs? What sort of proof would I need to discard my deeply held views?

If you can’t find any reason why you should think differently, then in essence you’re admitting that your views are purely visceral rather than evidence-based. Either change the basis of what you believe — i.e. try to find evidence for it, and discard it in the absence of said evidence — or admit that your beliefs have nothing to do with reality or rationality, but are instead the result of unthinking emotional or psychological attachment.

If we’re going to take up a belief or opinion based on “gut instinct,” faith, or whatever else you want to call it, then we might as well be honest, recognize it, and not hold it against others if they don’t see eye to eye with our views (after all, if said belief is based on personal feelings, rather than something objectively measurable, then you can’t expect everyone else to agree).

Also, there’s no harm in saying “I don’t know” or “I believe this based only on what I know.” It’s honest and it represents a fact of life: not everything is knowable to everyone.

To the best of my ability I try to hold myself to these standards. Otherwise, I leave it to others to call me out.

The ramifications of a book like this are very concerning. Not only could individual children die, but if enough people remain un-vaccinated, than herd immunity could weaken, meaning the disease could can momentum among the wider public. I’m glad it hasn’t received too much support.

Why Evolution Is True

Talk about poisoning the well: there’s a new book out by Stephanie Messenger called Melanie’s Marvelous Measles.  It’s an anxi-vaxer book for kids! The Amazon page describes the contents:

Melanie’s Marvelous Measles was written to educate children on the benefits of having measles and how you can heal from them naturally and successfully. Often today, we are being bombarded with messages from vested interests to fear all diseases in order for someone to sell some potion or vaccine, when, in fact, history shows that in industrialized countries, these diseases are quite benign and, according to natural health sources, beneficial to the body. Having raised three children vaccine-free and childhood disease-free, I have experienced many times when my children’s vaccinated peers succumb to the childhood diseases they were vaccinated against. Surprisingly, there were times when my unvaccinated children were blamed for their peers’ sickness. Something which is just not possible…

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Love, Knowledge, and Compassion

Humankind has become so much one family that we cannot insure our own prosperity except by insuring that of everyone else. If you wish to be happy yourself, you must resign yourself to seeing others also happy.

Bertrand Russell, a philosopher, social critic, historian, and logician, has long been one of the most influential people in my life. A noted humanist, atheist, and rationalist, he is the model for my own aspirations: to value love, knowledge, and compassion as the greatest pursuits in one’s life. Indeed, my personal mission statement, and that for this blog, is based upon these principles, which he so eloquently espouses in the following tract:

Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind. These passions, like great winds, have blown me hither and thither, in a wayward course, over a great ocean of anguish, reaching to the very verge of despair.

I have sought love, first, because it brings ecstasy – ecstasy so great that I would often have sacrificed all the rest of life for a few hours of this joy. I have sought it, next, because it relieves loneliness–that terrible loneliness in which one shivering consciousness looks over the rim of the world into the cold unfathomable lifeless abyss. I have sought it finally, because in the union of love I have seen, in a mystic miniature, the prefiguring vision of the heaven that saints and poets have imagined. This is what I sought, and though it might seem too good for human life, this is what–at last–I have found.

With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway above the flux. A little of this, but not much, I have achieved.

Love and knowledge, so far as they were possible, led upward toward the heavens. But always pity brought me back to earth. Echoes of cries of pain reverberate in my heart. Children in famine, victims tortured by oppressors, helpless old people a burden to their sons, and the whole world of loneliness, poverty, and pain make a mockery of what human life should be. I long to alleviate this evil, but I cannot, and I too suffer.

This has been my life. I have found it worth living, and would gladly live it again if the chance were offered me.

I should hope to live a life so rich in goodness and enlightenment. Few men have ever been as ethical and moral. Furthermore, Russell gives lie to the popular notion that a rational mindset devoid of religious belief is too cold, calculating, or prone to nihilism to be compassionate and sympathetic. Empathy, love, and the sincere desire to see other people be happy and prosperous are not predicated on any particular dogma – such things should pervade all of humanity, regardless of religious, political, or ideological persuasions. Virtue for virtue’s sake.

I could devote an entire blog on the prodigious amount of writing and thought that emerged from this great thinker, and I’ll no doubt be revisiting him more than once in the future. If anyone would like at least a sample of his wisdom, visit this collection of his sayings and observations.

Five Common “Logical Fallacies” Courtesy of Cracked

Believe it or not,, a humor site based off the defunct magazine, is actually a pretty good source of knowledge. It uniquely combines silly and even crass humor with some useful insights regarding topics in history, science,  politics, and other academic subjects.

Because of it’s funny, lighthearted delivery, as well as the brevity of its articles, you could easily learn a dozen new things within a single hour (I say this speaking from experience). I must admit that it could be very addicting and time consuming, especially as each page yields links to a half-a-dozen more fascinating subjects.

The one I’ve chosen concerns “Logical Fallacies” – I use quotes because they are technically not fallacies per se, but nonetheless are still prevalent myths and misconceptions that need good debunking. It definitely makes a lot of good points concerning our baseless assumptions and the pervasiveness of bad logic. You don’t have to be a science or philosophy buff to appreciate how crucial it is to learn from that.

I encourage you all to check out other articles as well, as they are similarly engaging and though-provoking (despite the veneer of inanity that the writers seem to live up). Granted, Cracked isn’t an academic source by any stretch – though it often includes hyperlinks and citations – but it’s a great place to get a quick dose of knowledge expressed in a fun and funny way.

I think it’s great to see knowledge proliferate in many forms. How you convey something is just as crucial as the subject itself, and knowledge is arguably worthless if it’s not communicable. Quick, irreverent approaches may work better than dry, high-minded ones. As long as people learn, the most important job is done.

Critical Thinking

The ability to think critically is perhaps the most important skill that any person could utilize. Critical thinking is applicable to almost any profession, scenario, or lifestyle – all of us inevitably find an instance or situation in which higher reasoning is vital. It can solve problems and disputes both external and internal. It helps us pick the right course of action to take on any given by, be it a career choice, a plan of study, or even relatively mundane decisions such as what to do on a given day.

Critical thinking is particularly relevant in the present day, as we’ve become saturated with knowledge and overloaded with too many claims and positions to sift through. Determining who to believe, which source to trust, and what argument is valid is difficult enough without being constantly bombarded with a stream of raw data. Thinking well allows us to discern flawed arguments from good ones, filter out sources that can be identified as unreliable, and allow us to properly weight various claims and assertions so as to reach a decision based on reason and logic.

Such thinking is also imperative for the proper functioning of democracy: people need to know the issues, policies, and political rhetoric that could make or break their nation’s future. We need to understand the implications of certain proposals, be able to tell the valid political platforms from the untenable ones, and make informed decisions about the choices we make at the ballot. With the current era of pervasive stagnation and polarization, critical thinking remains even more important: reasonable people are needed to transcend the vitriol and focus on the pertinent issues and what makes for good policy.

Finally, critical thinking is empowering. It inoculates you from hucksterism and manipulation, and makes you resistant to dishonest persuasion  or flawed arguments. It ensures that you understand the world better, and allows you to build a stronger foundation for your beliefs, practices, and political positions.

Of course, thinking critically is easier said than done. Anybody and everybody could claim to do it – indeed they often do so, as you hardly meet anyone who admits to not being a critical thinker. I like to think I’m a critical thinker, or at least as best as I can be. I’m sure I lapse at it just like anyone would. But for the most part, I try.

Critical thinking isn’t foolproof either: you could still fall for bad rhetoric, believe nonsensical things, and become manipulated. I don’t want to give the impression that it is a magic bullet against any and all though-provoking obstacles one may face. But it could certainly help, and having some inkling of it is better than none.

Thankfully, there are plenty of good resources out there to help us develop our higher mental faculties. Among my favorite is a site known as the Critical Thinker Academy, which includes the Critical Thinker Podcast. These resources are concise, easy to understand, and cover all the basics (and far more, if you’re willing to pay for membership).

There is also a site run by two professors and based out of Hong Kong that covers the topic rather well. It includes a lot of tutorials, practical scenarios to follow, and a glossary of related terms (recognizing various logical fallacies or what the different forms of argumentation are).

Finally, among my favorite reference is an old but timeless piece from the Skeptical Inquirer, A Field Guide to Critical Thinking. It sets out a list of guidelines and principles that are the foundations of good reasoning. It makes for a great and relatively short “How To” guide to thinking well. I definitely recommend a quick look at the very least.

Indeed, this list is hardly exhaustive. There is a lot of good material out there, just in the web alone, about how to be a better thinker. Ultimately though, what’s most important isn’t merely the capacity to think critically, but the desire to even care  in the first place. Without a commitment to truth, honesty, and knowledge, there is little reason to concern one’s self with the tools of enlightenment. The thirst for knowledge is the only thing more valuable than critical thinking, for the latter could not exist without the former; though the former cannot get far without the latter.

Logic, Reason, and Emotions

We readily assume that rationality is necessarily contingent upon our ability to suppress or suspend our emotions. In other words, we view emotion as an enemy of reason, and believe that both are mutually exclusive. To be more reasonable means to be more calculating, emotionless, and detached; to be emotional is to be erratic, reckless, and unstable.

This formula is pervasive across popular culture, entertainment, and media. The individual that exercises logic, critical analysis, or some other form of higher intelligence is usually portrayed as having suspended all feelings, to the point of being cold and aloof. By contrast, the more passionate and expressive character is often hot-headed, unreasonable, feckless. They’re two sides of the same coin, complementary in some cases but otherwise intrinsically in conflict.

This Manichean perception is widely held among most people and is arguably intuitive. In fact, I once prescribed to it as well, and only as I delved into topics like politics, science, philosophy, and other disciplines (mostly in the humanities) did I come to realize the beneficial synergy of my emotional proclivities and my intellectual ones. I now find myself battling this stigma that holds that any demonstration of logical or rational thinking is indicative of callousness or arrogance.

Many people even regard the use of inquiry, skepticism, and analysis as symptoms of an unethical and immoral worldview; those who take such approaches are stereotyped as mechanical utilitarians looking strictly at the efficiency or cost-benefit nexus of a given issue, dismissing the “human” element (though such methods of analysis aren’t necessarily heartless in and of themselves, but rather only when applied without ethical considerations). Maybe it’s just a confirmation bias on my part, but I’ve encountered it frequently enough in both media and personal experience to find it to be a relatively widespread notion.

For one, there is an issue of semantics. Emotions need not be personified strictly in their most explicit forms: being emotional isn’t only about being sensitive, quick to anger, or manic, as the average person would think. Emotions pervade every thought, action, and belief. When we’re saddened or perturbed by poverty, for example, we’re in a technical sense being emotional. If we feel a strong sense of justice and fairness, then we’re demonstrating an emotional investment: contentment at seeing justice prevail, compassion at making sure it does so, and anger if it’s violated. In both these examples, we’re also displaying the crucial emotion of empathy, which I’ll get back later.

Moreover, there is psychologically no such thing as distinct, inseparable components of either emotion or reason. No normal human being could ever completely shut down or suppress one or the other, nor would doing so  strengthen one at the expense of the other.  Displaying one’s emotions and exercising one’s higher thinking faculties is not necessarily a zero-sum game. Certainly, we all have varying degrees with which we utilize either one, depending on the subject, mood, or individual (as well as numerous other externalities that shape our minds, such as chemical imbalances, how we were raised, neurological pathologies, and so on).

In reality, the two areas are inseparable, and often even dependent upon one another. We need an emotional capacity for our rational minds to operate. Without emotions, we have no deep-seated sense of right and wrong. We’re naturally capable of being passionate about truth, justice, the well-being of others, and other moral and ethical considerations. Therefore, we need these emotional considerations to feed our desire to think critically and determine what is right from wrong, or to figure out what course of action is best for benefiting ourselves, our loved ones, and our sense for virtue.

Too often, we assume being logical and rational means you must be cold, mechanical, and even inhumane. On the contrary, a more intellectually and philosophically developed mind is far more suited to developing a reliable basis for justice and morality, provided that our heart is also in the right place. Generally speaking, however, a well-developed and critically-thinking mind is more adept at weighing the many options of a given choice, or of knowing the deeper details of topics pertaining to law, politics, psychology, and other crucial areas dealing with human well-being in some form or another.

Thus, exercising our higher faculties requires a level of emotional commitment. There is no reason to think critically or rationally about the nature of injustice or the solutions for poverty without some emotional investment – compassion, altruism, disgust with unfairness – to promote it in the first place.  If I don’t care about living a virtuous life, or helping others to flourish, why think about it in the first place?

This is not to suggest that one must be a sage to be capable of just or good behavior. Plenty of smart people can be well-versed in absorbing the raw data of knowledge, but be less keen in their ability to perform the requisite critical thinking needed for understanding and developing a moral/ethical worldview; similarly, they may also have less empathy, or have failed to apply their higher faculties for good causes.

Thus, the key to being a good person – defined here as being knowledgeable, virtuous, and compassionate* – is to dispel this false choice between being emotional and being rational. Sure, lacking self-control of one’s emotions can be detrimental, just as thinking too much can lead to indecision or detachment from a subject. But that doesn’t mean that being emotional or reasonable is, in principle, a matter of choosing one and rejecting the other. Humans are meant to display emotions. It’s healthy and necessary, and it makes life better for us. But we’re also meant to be problem solvers, to use our unique capacity to think, analyze, and reason to address any number of obstacles – practical, ethical, existential – that inevitably come our way.

As a great philosopher once said, an education of the mind without an education of the heart is no education at all. Indeed, the same works the other way around. We must utilize the best aspects of our minds, and take a holistic and balanced approach to how we better ourselves, our loved ones, and the world around us.

Qualia Soup

Once again, I  am pleased to have found a good dose of wisdom on YouTube, courtesy of a user known as Qualia Soup. He posts a series of Flash-animated videos that crisply address several scientific and philosophical topics, particularly arguments in favor of freethought, reason, and skeptical inquiry; and a critical dissection of religious dogma, illogical thinking, and pseudo-scientific ideas such as intelligent design.

Personally, I find the videos to be of great quality and are well presented, with good narration and structure (his scholarly English accent certainly helps). What I like most is the way he addresses his given topic in a clear and concise manner, dealing with common misconceptions and presenting the material in a way that engages those viewers with little understanding of the topic without being patronizing. I respect anyone that can find a balance between properly explaining a subject and doing so in a way that is easy to understand (especially when it comes to relatively difficult topics such as logic, evolution, morality, and so on). My only (minor) criticism is his tendency to move a bit too fast sometimes.

The following are a few of my favorite, though I haven’t seen them all:




I encourage you to look into more of his work. I think even my religious friends could appreciate some of the interesting arguments he presents, and could no doubt agree with at least some of the irrational arguments and beliefs he invalidates.  At the very least, I hope this encourages some reflection and open-mindedness, whether you agree with the arguments or not. As always, my purpose is nothing more but to share certain philosophical and logical positions that appeal to me, either to enlighten others or to initiate a discourse (be it with me, my resources, or within your own mind).

Hope you all enjoy.