## Tag Archives: Math

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It’s often said that the distinction between deductive and inductive reasoning is that one argues from the general to the specific, and that the other does so from specific to the general, but this is not correct across all forms of these sorts of reasoning.

Each can work both ways. Whoa. That’s quite an assertion, so I’ll attempt to show why here…

In deductive reasoning, the truth of an argument’s conclusion automatically follows from the truth of its supporting statements if it’s valid. Also, it’s possible for one or more of such an argument’s supporting statements to be false, and this renders it unsound even when valid. Valid but knowingly unsound arguments are not persuasive, and cannot ethically be used as though they were compelling.

Deductive arguments reorganize what we know rather than providing any new data. Deductive conclusions cannot go beyond what’s expressed or implied in their supporting statements.

Inductive reasoning, such as some of the informal sort more typically used in everyday life, can only justify conclusions as more or less probable, depending on the strength of the argument and the prior adherence of an audience to its evidence. This form of reasoning does provide new knowledge, by moving us from the known to the unknown, unstated, and not implied.

In generalizing from particular examples, I’ll show how it may be deductive and then inductive:

If I were to be on the shoreline of my local beach, and noted that the pebbles found there were worn smooth and comfortable to the touch by the actions of water and sand, and were to completely and perfectly enumerate each and every such pebble on the shore, to find them all worn smooth, an unlikely and difficult task at best, I would know with certainty that all of these pebbles were smooth and comfortable to the touch. Each and every one. The argument would then follow deductively.

If, though, I were to find one smooth pebble, and then another, and so on, and after noting from a large enough but limited sample of such stones that they are almost all smooth to the touch, though I may find a few which are not, I could conclude inductively that they are more often smooth and worn than not. The argument follows to a high degree of probability based on the size and representativeness of the sample examined, and it is an acceptable substitute for the certainty we cannot usually get in measuring things in the real world.

The things to look out for when generalizing are known as the fallacy of composition, and the hasty generalization, these errors made when we attempt to apply deductive certainty where it does not belong, the first in assuming that the whole of a population is necessarily like the parts, and the second in drawing an unfounded general conclusion on the basis of too little sample data.

Now from general to specific, classifying rather than generalizing.

If I were to get a perfect count of all sand-ground stones on this hypothetical beach, not over-counting or skipping some, and they all were worn smooth, then I could conclude certainly that any one of these stones was going to be smooth in texture just like all the others. The argument would follow deductively.

But if I were to do the more likely thing, and count a fairly sizable number of these pebbles, all showing signs of wear and smoothness, then I could argue with a good chance of being correct in saying that any one of the stones I pick up would be ground and smooth. It would then be an inductive argument.

The error to avoid committing here is the very opposite of the fallacy of composition, the fallacy of division, in which one misapplies deductive certainty by claiming that the part is necessarily like the whole.

To close out, the examples I used in this post come from a quote attributed to Sir Isaac Newton, that cranky and brilliant English guy, which goes:

I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. ~ Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton (1855) by Sir David Brewster (Volume II. Ch. 27).

Cropped image of a Socrates bust for use in philosophy-related templates etc. Bust carved by by Victor Wager from a model by Paul Montford, University of Western Australia, Crawley, Western Australia. (Photo credit: Wikipedia)

This post was originally published in 2011, and since then I’ve decided to give it new life and clear up difficulties in the text. I decided to use it once for for the pilot entry of my current Project Logicality. I hope it adds to the online discussion of the virtues of reason despite the decidedly unreasonable tendencies of the human species.

What is it that I mean when I say ‘argument?’ When I use this term, I don’t mean quarrelsome bickering accompanied by yelling and screaming, nor do I mean a mere shadow of an argument where debaters try to undermine the legitimacy of each others’ position without attempting to reach a real understanding or settling anything.

When I say, “argument” I mean it in the context of any rational discussion with constructive intent, not an attempt to thwart constructive ends through fallacious means.

Here, I mean that the parties involved act to offer reasons, premises, rationales, and justifications for the statements, the claims, and the ideas that they put forth. They want others to accept these, not merely by pandering to their prejudices or appealing to their biases, nor upon the use of legal or physical force, but by winning the free assent of that audience — an audience treated as though it were in principle intelligent, educated, and capable of exercising rigorous critical judgement.

I refer to argument in the sense of modern argumentation theory, a vibrant field of study involving the making and use of messages to influence others, by appealing to their willingness to cooperate — this is essential for the conditions of a viable free society.

Any coherent social structure, especially a functioning representative democracy with a large number of people needs some means of mutual influence between its members, of and for the viability of its governing system, however imperfect its governing body in practice. Perfection in matters of human endeavor is a chimera.

Argumentation as a field of study crosses paths with three other areas of intellectual endeavor:

First, it converges with Logic, the broader study of the structures we use in all processes of reasoning — this includes formal logic, mathematical and symbolic logic where the conclusion of a valid argument is alleged to be certainly true if the premises used to support it are also true.

But argumentation concerns itself more with informal logic, the everyday reasoning we engage in within typical discussions — in which the statements we wish to support do not follow with certainty, but to a degree of probability depending on the strength of our reasons and the willingness of the audience to accept them.

In argumentation, even the very idea of certainty depends on the audience addressed.

Language is important in informal logic, because informal argumentation depends heavily on the use of language as the content of the argument. Language is more than merely decorative window-dressing for an informal argument, but an essential part of the argument’s inherent meaning. The language that an argument is cast in cannot be taken from the argument itself without rendering it sterile and empty.

Second, argumentation converges with Rhetoric, originally one of the seven Liberal Arts — it is more than just vacuous or bombastic and flowery ornamentation in speech as is commonly supposed, but in the technical sense it is the broader study of how people are influenced by messages.

It is from Rhetoric that argumentation gets the concern with the requirements of an audience — its needs, disposition, and outlook must be considered by the arguer in making their case.

Third, argumentation crosses over with dialectic, a term that many people still associate with the concept of an opposition between grand historical forces, like the opposition of capitalism and communism depicted in Marxist social theory.

This concept has a different meaning, and dates at least since the Socratic method,  given in the dialogues of Plato, and others, in which fictionalized persons are seen to engage in a sort of give and take exchange of questions and answers to resolve a dispute or reveal the truth of a matter.

This sort of questioning is similar to the use of cross-examination of witnesses in modern legal courts by the prosecuting attorneys in a case to uncover inconsistencies in testimony and to reveal possibly questionable motives.

Argumentation is the meeting point of all of these fields, and with it, we can clarify our understanding of our positions, resolve disputes, reach sound decisions for collective actions we may undertake, engage in formal and often productive debates, and, with ourselves as the audience, think through personal problems we may face or get out of a rut.

Argumentation as a process of giving reasons for our claims is one of the most important abilities we have as humans, and no matter our level of education, we can all benefit from the ability to arrive at better answers to questions and make more sound decisions than we otherwise might.

Argumentation isn’t just for egghead academics: Clear thinking and having good reasons for what we believe and do are for everyone. As humans, we are not always rational, but we have a sense of reason, one that once nurtured and practiced can serve us and enfranchise us as informed, effective, and smart voters very well indeed.

G’day. In this installment, I’ll show a few recent images made using custom parameter sets, courtesy of Mandelbulber.

Perhaps it’s just status quo bias on my part — it’s always good to acknowledge the possibility of one’s own biases, in my view, and even better to more fully understand when they might play into our decisions — I’ve long considered MB my favorite app for doing 3D images, at least for my laptop’s operating system.

This raises the question of whether there are any more recent apps that do the same thing, or better, for the hardware and OS I use, and I’ll be looking into that during the remainder of the week, which should be fun.

I’ve begun the draft for the first installment of a series of fractal generation tutorials (Yes, I’ll be posting it in several entries instead of one long post for ease of presentation.), and that project should be completed by next year when study resumes.

I’m hoping whatever holiday you celebrate this time of year is an absolute blast, and so, here are this week’s images:

Fluidity

Petrolithos

Photokronik

All JPEG, PNG & GIF images in this post are original works by the author,created by

way of XaoS, Mandelbulber, Fractal Domains, and Ultra Fractal and are copyright 2013 by Troy Loy.

Good evening, and happy Tyr’s day. Tonight I have some further explorations of Ultra Fractal‘s capabilities, and generating at least one of tonight’s images pushed the system to it’s limits, forcing me to reboot the ‘toy’ to clear memory for further use.

But enough foolishness. Here are tonight’s images…

Using a coloring algorithm to assist in generating this, it is an Apollonian gasket gone horribly wrong, that to me seems to resemble the cut-open insides of a piece of exotic fruit, while this next one…

…uses a configuration referred to as ‘Grandma’s Special,’ and it is very particular in the proper use of color settings. I think I got lucky with this one! There is a discussion on the Kleinian groups and some useful parameters for them for use in Ultra Fractal.

Finally, this is a slightly different take that ate up most of my ‘toy’s memory while generating it as the original larger image, but I think it was worth it. In future, I’ll have to adjust the settings a bit more carefully!

Lesson learned!

As the week progresses, the exploratory work gets more challenging, but also more fun. Let’s see how far I can push this!

Talotaa frang.

The net of the great god Indra was said to span all of space, finer than silk, and strung with pearls reflecting each other and all others along the infinite reach of the net at once forever.

Thus seemingly was foretold in Buddhist scripture the vision of Felix Klein, and the mathematical constructions used in the fractal type of this post, as paraphrased from the book, ‘Indra’s Pearls.’

Good evening, and welcome to the launch of the first weekday of this project’s posts, my attempt to more fully explore this sometimes cross but rewarding fractal type based on the mathematics of the interaction of spirals and the objects it leads to.

Today I’ve got four images to show, what I think to be the very best from today and last night generated by my current favorite app, Ultra Fractal over a period of several hours.

This on was made using my first parameter file created for this series (I generated ten such files, adjusted as needed), using an extremely well-designed coloring algorithm, ‘Double Cusp’ Kleinian group settinmgs and a rather fortuitous combination of color gradient settings.

This one uses a figure altered threefold with a kaleidoscopic transformation, using a different coloring algorithm and different but still striking gradient settings. I like to randomize and shift gradients around when coloring these, just to see what I’ll get, though the app certainly allow easy non-random color adjustments as well.

This one used my second file created, and used a different coloring algorithm than the previous two and different Kleinian groups (Schottky bubbles) to generate this image. Of today’s pieces, it took the longest to render, but was well-worth it I think.

The scope and potential of this fractal type is amazing, despite the relatively simple starting idea of interacting spirals, or perhaps even because of it. I’ll include further images tomorrow, and may you all fare well the remainder of this week.

Talotaa frang.

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