Proof and Reasoning Concepts
- Reasoning begins with examination, comparison, and evaluation.
- Examination can include discovering similarities and differences to organize, categorize, seriate, make connections, create correspondences and create other structures. All which can be connected for deeper understanding and to solve problems.
- Organizing similar problems with their solutions may provide examples to use to generalize patterns, explanations, expressions to describe conditionw in which the mathematical ideas are valid.
- When expressions, equations, or procedures are deemed valid in enough specific situations, usually the next step is to see if the method is valid for all instances (infinite).
- A procedure or equation may be verified or proven if a reason can be given for every possible solution.
- Conjecture is a statement, which may be proven true or false. It may be based on an idea or conclusion with insufficient evidence or reason to prove. As such it can be a starting point for a proof.
- One counter example will disprove a conjecture.
- When explaining a proof a reason needs to be given for every change made.
- Pictures and charts can be used to help support reasoning and proof.
- Numbers can be compared by Numbers can be made equal by thinking of their values on a balance (equal arm balance) and changing the numbers on either side until it is balanced.
- Changes to those numbers are created with the mathematical operations.
Big Ideas for unpacking and mapping - Source Science for All Americans
Some aspects of reasoning have clear logical rules, others have only guidelines, and still others have almost unlimited room for creativity (and, of course, error). A convincing argument requires both true statements and valid connections among them. Yet formal logic concerns the validity of the connections among statements, not whether the statements are actually true. It is logically correct to argue that if all birds can fly and penguins are birds, then penguins can fly. But the conclusion is not true unless the premises are true: Do all birds really fly, and are penguins really birds? Examination of the truth of premises is as important to good reasoning as the logic that operates on them is. In this case, because the logic is correct but the conclusion is false (penguins cannot fly), one or both of the premises must be false (not all birds can fly, and/or penguins are not birds).
Very complex logical arguments can be built from a small number of logical steps, which hang on precise use of the basic terms "if," "and," "or," and "not." For example, medical diagnosis involves branching chains of logic such as "If the patient has disease X or disease Y and also has laboratory result B, but does not have a history of C, then he or she should get treatment D." Such logical problem solving may require expert knowledge of many relationships, access to much data to feed into the relationships, and skill in deducing branching chains of logical operations. Because computers can store and retrieve large numbers of relationships and data and can rapidly perform long series of logical steps, they are being used increasingly to help experts solve complex problems that would otherwise be very difficult or impossible to solve. Not all logical problems, however, can be solved by computers.
Logical connections can easily be distorted. For example, the proposition that all birds can fly does not imply logically that all creatures that can fly are birds. As obvious as this simple example may seem, distortion often occurs, particularly in emotionally charged situations. For example: All guilty prisoners refuse to testify against themselves; prisoner Smith refuses to testify against himself; therefore, prisoner Smith is guilty.
Distortions in logic often result from not distinguishing between necessary conditions and sufficient conditions. A condition that is necessary for a consequence is always required but may not be enough in itself—being a U.S. citizen is necessary to be elected president, for example, but not sufficient. A condition that is sufficient for a consequence is enough by itself, but there may be other ways to arrive at the same consequence—winning the state lottery is sufficient for becoming rich, but there are other ways. A condition, however, may be both necessary and sufficient; for example, receiving a majority of the electoral vote is both necessary for becoming president and sufficient for doing so, because it is the only way.
Logic has limited usefulness in finding solutions to many problems. Outside of abstract models, we often cannot establish with confidence either the truth of the premises or the logical connections between them. Precise logic requires that we can make declarations such as "If X is true, then Y is true also" (a barking dog does not bite), and "X is true" (Spot barks). Typically, however, all we know is that "if X is true, then Y is often true also" (a barking dog usually does not bite) and "X seems to be approximately true a lot of the time" (Spot usually barks). Commonly, therefore, strict logic has to be replaced by probabilities or other kinds of reasoning that lead to much less certain results—for example, to the claim that on average, rain will fall before evening on 70 percent of days that have morning weather conditions similar to today's.
If we apply logical deduction to a general rule (all feathered creatures fly), we can produce a conclusion about a particular instance or class of instances (penguins fly). But where do the general rules come from? Often they are generalizations made from observations—finding a number of similar instances and guessing that what is true of them is true of all their class ("every feathered creature I have seen can fly, so perhaps all can"). Or a general rule may spring from the imagination, by no traceable means, with the hope that some observable aspects of phenomena can be shown to follow logically from it (example: "What if it were true that the sun is the center of motion for all the planets, including the earth? Could such a system produce the apparent motions in the sky?").
Once a general rule has been hypothesized, by whatever means, logic serves in checking its validity. If a contrary instance is found (a feathered creature that cannot fly), the hypothesis is not true. On the other hand, the only way to prove logically that a general hypothesis about a class is true is to examine all possible instances (all birds), which is difficult in practice and sometimes impossible even in principle. So it is usually much easier to prove general hypotheses to be logically false than to prove them to be true. Computers now sometimes make it possible to demonstrate the truth of questionable mathematical generalizations convincingly, even if not to prove them, by testing enormous numbers of particular cases.
Science can use deductive logic if general principles about phenomena have been hypothesized, but such logic cannot lead to those general principles. Scientific principles are usually arrived at by generalizing from a limited number of experiences—for instance, if all observed feathered creatures hatch from eggs, then perhaps all feathered creatures do. This is a very important kind of reasoning even if the number of observations is small (for example, being burned once by fire may be enough to make someone wary of fire for life). However, our natural tendency to generalize can also lead us astray. Getting sick the day after breaking a mirror may be enough to make someone afraid of broken mirrors for life. On a more sophisticated level, finding that several patients having the same symptoms recover after using a new drug may lead a doctor to generalize that all similar patients will recover by using it, even though recovery might have occurred just by chance.
The human tendency to generalize has some subtle aspects. Once formed, generalities tend to influence people's perception and interpretation of events. Having made the generalization that the drug will help all patients having certain symptoms, for example, the doctor may be likely to interpret a patient's condition as having improved after taking the drug, even if that is doubtful. To prevent such biases in research, scientists commonly use a "blind" procedure in which the person observing or interpreting results is not the same person who controls the conditions (for instance, the doctor who judges the patient's condition does not know what specific treatment that patient received).
Much of reasoning, and perhaps most of creative thought, involves not only logic but analogies. When one situation seems to resemble another in some way, we may believe that it resembles it in other ways too. For example, light spreads away from a source much as water waves spread from a disturbance, so perhaps light acts like water waves in other ways, such as producing interference patterns where waves cross (it does). Or, the sun is like a fire in that it produces heat and light, so perhaps it too involves burning fuel (in fact, it does not). The important point is that reasoning by analogy can suggest conclusions, but it can never prove them to be true.