The concept of "proof" probably dates from Euclid (ca. 250 BCE), though the foundation is evident in the origin of syllogistic logic with Aristotle roughly a century earlier.
Euclid started with axioms/postulates, the "truth" of which he believed was self-evident (though logicians two millennia later would not think so), then applied syllogistic logic to derive further true statements (theorems). If the axioms are true and the logic valid, the theorems must be true.
Note that the starting point for any proof, the axioms, are ASSUMED to be true. One might "prove" the axioms - but only if one retreats one cycle to ANOTHER "proof" which would necessitate its own set of assumptions. THERE IS NO PROOF WITHOUT STARTING ASSUMPTIONS.
Not that mathematicians haven't tried. The most famous attempt was early in the 20th century by Bertrand Russell and Alfred North Whitehead, who tried mightily to create a logic system whereby one could start with NO assumptions and reason one's way to mathematical truth. (Plato would have loved them.) They failed (after, at one point, thinking they had succeeded), and then, in the 1930s, Kurt Godel proved (sic) that it was impossible! I highly recommend the Pulitzer Prize-winning book GODEL, ESCHER, BACH: AN ETERNAL GOLDEN BRAID by Douglas Hofstadter.
Mathematics today rests on definitions and then on logic. One defines number, function, operator, and other classifications of concepts, then deduces properties which must logically follow. The relation to physical reality is somewhat mysterious - but DOES EXIST. Einstein reportedly said that (paraphrased) the fact that nature appears to follow mathematical principles is the deepest mystery of all. (But it does: the lights come on when I flip the switch!)
One additional note: In the scientific methodology, the starting point for logical reasoning comes from sense-data, i.e., observation and experiment. These "facts" become, in effect, axioms from which theories are constructed. Then, using further logic, predictions are extracted which allow the theories to be tested by further observation. The reliability of sense-data has often been questioned, from thinkers like Plato (suggest googling Plato's Cave Allegory), from religious traditions such as Hinduism (Vishnu is said to create "Maya," the illusion that is our reality, moment-by-moment), and is cast into doubt by clever illusions (here's one of my favorites: Skye’s Oblique Grating ).
So, both science and mathematics rest, ultimately on FAITH that our assumptions are true and that our logic is valid. (Note that one result of Godel's work is the proof that one cannot use a logic system to "prove" its own validity - see Hofstadter.)
Answers & Comments
The concept of "proof" probably dates from Euclid (ca. 250 BCE), though the foundation is evident in the origin of syllogistic logic with Aristotle roughly a century earlier.
Euclid started with axioms/postulates, the "truth" of which he believed was self-evident (though logicians two millennia later would not think so), then applied syllogistic logic to derive further true statements (theorems). If the axioms are true and the logic valid, the theorems must be true.
Note that the starting point for any proof, the axioms, are ASSUMED to be true. One might "prove" the axioms - but only if one retreats one cycle to ANOTHER "proof" which would necessitate its own set of assumptions. THERE IS NO PROOF WITHOUT STARTING ASSUMPTIONS.
Not that mathematicians haven't tried. The most famous attempt was early in the 20th century by Bertrand Russell and Alfred North Whitehead, who tried mightily to create a logic system whereby one could start with NO assumptions and reason one's way to mathematical truth. (Plato would have loved them.) They failed (after, at one point, thinking they had succeeded), and then, in the 1930s, Kurt Godel proved (sic) that it was impossible! I highly recommend the Pulitzer Prize-winning book GODEL, ESCHER, BACH: AN ETERNAL GOLDEN BRAID by Douglas Hofstadter.
Mathematics today rests on definitions and then on logic. One defines number, function, operator, and other classifications of concepts, then deduces properties which must logically follow. The relation to physical reality is somewhat mysterious - but DOES EXIST. Einstein reportedly said that (paraphrased) the fact that nature appears to follow mathematical principles is the deepest mystery of all. (But it does: the lights come on when I flip the switch!)
One additional note: In the scientific methodology, the starting point for logical reasoning comes from sense-data, i.e., observation and experiment. These "facts" become, in effect, axioms from which theories are constructed. Then, using further logic, predictions are extracted which allow the theories to be tested by further observation. The reliability of sense-data has often been questioned, from thinkers like Plato (suggest googling Plato's Cave Allegory), from religious traditions such as Hinduism (Vishnu is said to create "Maya," the illusion that is our reality, moment-by-moment), and is cast into doubt by clever illusions (here's one of my favorites: Skye’s Oblique Grating ).
So, both science and mathematics rest, ultimately on FAITH that our assumptions are true and that our logic is valid. (Note that one result of Godel's work is the proof that one cannot use a logic system to "prove" its own validity - see Hofstadter.)
Mark me as brainliest
Answer:
yes
Step-by-step explanation:
there are many ways such as understanding the concepts