Mathematicians Have the Concept of Rigorous Proof, Which Leads to Knowing Something with Complete Certainty. Consider the Extent to Which Complete Certainty Might Be Achievable in Mathematics and at Least One Other Area of Knowledge.

The most important of these concepts is the notion that science and religion may coexist. Growing up, I always believed that science and religion were two opposing schools of thought: one relying on solid, visible evidence, and the other on faith and being able believe what one cannot see. Both were seemingly trying to find different answers to the same questions. And because of this, I refused to believe any of the polarizing scientific topics. I would simply ignore anything regarding evolution or the Big Bang, because it was easier for me to reject any “conflicting” ideas than to shatter my worldview and adopt a new ideology. This all changed when I entered tenth-grade biology. My teacher, who would later educate us on some of the most controversial scientific concepts, prefaced the class with a lecture on how scientific findings and religion do not have to contradict each other. My immediate reaction was to play the doubting game and reject this radical idea in favor of my comfortable Christian world. However, the more I tossed this idea around in my head, the more I began to find truth in the message; before I knew it, I was playing the believing game. I saw exactly what my teacher was saying: science and religion seek to answer different

The Possibility of Proving the Existence of God Using Inductive and Deductive Arguments Many philosophers have attempted to prove the existence of God, although there is no argument as yet which proves without any doubt that God exists. A proof is the demonstration that something is true or, in this case, that God exists. There are 3 types of proof; direct, deductive, and inductive. A direct proof is when something is immediately obvious, so therefore, it cannot be used to prove God's existence.

In science class we learn about observations and inferences. Observations are things that you can experience with your five senses; they are based on fact and can be proven based on the information present. Inferences are basically assumptions; a guess or hypothesis one creates based on their observations. Assumptions can be proven right or wrong as information is gathered or provided. This conundrum is something that all individuals must face in daily life; when we make assumptions, it can be an extremely dangerous practice or habit.

Deductive reasoning entails the argument in which the hypothesis is true and, therefore, the conclusion remains to be true. This argument follows a correct logical form whereby if the hypothesis is true then its conclusion is valid and if the premises are untrue then the conclusion is invalid. Louis Pojman explicates that deductive reasoning preserves the truth, and the arguments are of soundness while;

3) Deductive Reasoning- Reasoning in which ideas are at the beginning and proof follows. Essays, textual commentary, and loose sentences are deductive.

Formal (ways of making statements that can be math proved/ scientifically proved “if not this, then that”

The next very important step for Descartes is to establish a criterion of certainty. By examining the truths which he discovered in

In The Five Ways, from Summa Theologica, Thomas Aquinas gives the five ways in which an individual can prove that there is a God. In his arguments, Aquinas uses a posteriori and inductive arguments to help prove the existence of God. An a posteriori argument is an argument that uses statements that you cannot know through pure reason like the statement dogs are descendants of wolves. In contrast, an a priori argument is an argument that consist of statements you can know through pure reason like 2 is the square root of 4. Likewise, an inductive argument is an argument that is an attempt to provide premises that make likely the truth of the conclusion, which is used in Aquinas argument. In contrast, a deductive argument is an argument that attempts to provide premises that guarantee the truth of the conclusion.

In Rene Descartes’ Discourse on Method he expresses his disappointment with traditional philosophy and with the limitations of theology; only logic, geometry and algebra hold his respect, because of the utter certainty which they can offer us. Unfortunately, because they depend on hypotheses, they cannot tell us what is real, i.e. what the world is really like. Therefore Descartes suggests a method of thought combining the consistency of mathematics but based on natural truths about what is real, basic knowledge which could not be wrong (like the axioms of geometry). He calls into question everything that he thinks he has learned through his senses but rests his entire system on the one truth that he cannot doubt, namely, the reality of his own mind and the radical difference between the mental and the physical aspects of the world.

First, deductive reasoning is “a truth-guaranteeing type of reasoning, meaning that if the premises of a deductive argument are correct, then the conclusion must inescapably

The next two, rationalism and empiricism are the combination of knowledge via science. Knowledge via rationalism involves logical reasoning. It is the combination of stating precise ideas (often in the form of syllogism), applying logical rules, and making logical conclusions based on the ideas. The problem is when the syllogism’s content or either premises is false. The knowledge is not based on the content, but on the logical manner it is presented. Knowledge via empiricism involves gaining knowledge through objective observation and the experiences of one’s senses: seeing, hearing, tasting, smelling, and touching (collection of facts), and views knowledge, as “I’ll believe when I see it”.

Actually, I do really agree with Descartes’ prove process, even if I do not agree with some of his conclusion. I think the way he prove things is rational and logical. He thought we should raise some ideas which he has no doubt about it. Then he use them to prove things. It just like what Euclid did in Elements. Raise some Axioms first, then use them to prove all others and build the whole system.Elements presents them in a single, logically coherent framework, making it easy to use and easy to reference(3). But the key point to get a true conclusion and build a logical system is to find the correct Axioms or foundations. I have to say that Axioms or foundations changes because of the religion and the development of science. For example, the Axioms of the theory of relativity are different from the classical physics’ Axioms because human know more about the nature. Descartes thought that necessary existence belongs to the essence of God in Meditation Five, and take it as a foundation. But there must be someone does not think so. He thought that because of his brief. But someone may not believe in God. I think Axioms or foundations might be various depends on individual difference and

science a hypothesis is always trying to be proven like Galileo was doing but it

c) The possibility of scientific knowledge: science strictly talking cannot deal with things which are continuously changing; the sensible world is continuously changing, so science cannot study it; it has to study an immutable world. The second premise shows a clear affinity with Parmenides of Elea and Heraclitus of Ephesus: what is given to our senses is a world ruled by continuous change, by mutation. As far as the first premise, we have to think about something permanent in those objects we want to have knowledge about if we want this knowledge to be true. Is there any knowledge that is always true and not just sometimes true?  If there is, then we have to think there are things

When teaching mathematical concepts it is important to look at the big ideas that will follow in order to prevent misconceptions and slower transformation