A2A

Yes.

Reasoning in logical steps and not accepting leaps which violate the rules of math can be applied to other sub and objects and other lines of reasoning.

A2A

Yes.

Reasoning in logical steps and not accepting leaps which violate the rules of math can be applied to other sub and objects and other lines of reasoning.

I think so.

When I studied mathematics in college I also also studied logic, both formal and symbolic. Logic is a key element of mathematical proof, and mathematical reasoning employs the same logical skills as other problems, only dealing with different objects.

The most useful thing I learned in math was rigor, a discipline of insuring that every assumption is documented and that all cases are accounted for.

I don't know how useful advice to just "practice" or "study" is, because very few people are capable of productively investing huge amounts of time in something they don't care about. To learn math you need to find a problem you actually want to think about and then just start exploring it.

Here are some various questions of different natures you might consider:

- A raw audio file that lasts five minutes might take nearly 100 MB on disk. Compressed into an mp3 file it only takes a tiny fraction of that. How does this work? Can you write your own mp3 encoder/decoder/player from scratch?
- By exami

I don't know how useful advice to just "practice" or "study" is, because very few people are capable of productively investing huge amounts of time in something they don't care about. To learn math you need to find a problem you actually want to think about and then just start exploring it.

Here are some various questions of different natures you might consider:

- A raw audio file that lasts five minutes might take nearly 100 MB on disk. Compressed into an mp3 file it only takes a tiny fraction of that. How does this work? Can you write your own mp3 encoder/decoder/player from scratch?
- By examining planetary data, Kepler was able to make some fairly bizarre-sounding observations about the orbits of planets: for instance, the line joining a planet to the sun sweeps out equal areas in equal units of time. Isaac Newton was able to use these observations to demonstrate that planetary orbits are just consequences of the same force of gravity that's familiar from everyday life on earth. How?
- If we think about the various ways of tiling a plane with identical tiles in a repeating pattern, there are various ways of doing it: you can use a simple checkerboard-type pattern, or you can mix it up by using a checkerboard where you rotate the tiles, or you can use a honeycomb-like hexagonal pattern, and so forth. As it turns out, there are exactly 17 ways of doing this. Why? What about ways of tiling a plane with identical tiles in an irregular pattern?
- In a similar vein, there are exactly five platonic solids: the tetrahedron, cube, octahedron, icosahedron, and dodecahedron. Why are there exactly five? What if we relax some of the requirements on a platonic solid -- what then?
- How does Google sort through billions of webpages in a couple of seconds and find pages that are both of high quality and directly relevant to your search?
- Consider polynomial equations in two variables, like x4+y4=x2x4+y4=x2 or y2=x3y2=x3. These represent curves in the plane. What can be said about these curves?

Obviously this isn't an exhaustive list; it's just there to give you some ideas.

Logical reasoning is an intellectual ability which includes abstraction and often requires imagination and mental courage (open mind) to think about what if -situations. Most mathematical skills also need this ability to use your reason abstractly, but they can also involve some other kind of thinking. For example, in geometry it is good to have an ability to think about objects in space and to mentally see 3-dimensial objects from different angles.

Formal logic is usually learned rather late in school because you have to have the capacity of abstraction, making formal operations and following

Logical reasoning is an intellectual ability which includes abstraction and often requires imagination and mental courage (open mind) to think about what if -situations. Most mathematical skills also need this ability to use your reason abstractly, but they can also involve some other kind of thinking. For example, in geometry it is good to have an ability to think about objects in space and to mentally see 3-dimensial objects from different angles.

Formal logic is usually learned rather late in school because you have to have the capacity of abstraction, making formal operations and following the rules literally. Most children have learned to do something with numbers by this age and laymen call this counting mathematics. Some of these kids have not been very good in these early exercises and they may have math anxiety: they are too anxious to perform. The same children can become decent at logical thinking later, but their basic understanding of mathematical thinking can be too limited for them to learn new things. And the anxiety is still their, constant failing does not help in erasing it.

In the brain there are separate areas for understanding number concepts. Some people have difficulties in undertanding even the relationships between smaller and bigger digits and this kind of brain makes it hard to do any algebra. Those kids and adults still can perform in logical tasks. Therefore, at least some parts of thinking is not the same.

There are IQ tests which claim to test logical and mathematical reasoning but test only a part of the skills required in those areas. Algorithmic thinking might be a better test for abilities required in both logic and mathematics than mental rotation tests, especially because mental rotation tests may cause test anxiety and some people underperform in them.

It's a lot of work. First you separate what is similar from what is different. Take Landsteiner. First he got samples of blood from several colleagues. He separated each sample into clear serum and red blood components. Then he mixed each sample with all the others. Next he constructed a grid of where he found the clumping reactions. That was his pattern. That led to categorization of blood into types. Widening the size of the inductive sample lead to discrimination of further types. Today, there are hundreds of types that have been isolated. But it required dozens of further inductions, in c

It's a lot of work. First you separate what is similar from what is different. Take Landsteiner. First he got samples of blood from several colleagues. He separated each sample into clear serum and red blood components. Then he mixed each sample with all the others. Next he constructed a grid of where he found the clumping reactions. That was his pattern. That led to categorization of blood into types. Widening the size of the inductive sample lead to discrimination of further types. Today, there are hundreds of types that have been isolated. But it required dozens of further inductions, in cytology, protein structure, etc. and what Whewell would call the "consillience of inductions" to determine a theory (antibodies) that explained the clumping reaction causally. Even to describe all the details of how it was experimentally determined that cells have proteins in their walls, and that the serum has other proteins that interlock with them, and all the other details would probably take a full book. By the way, it would be a great book. But there is no such book. (People don't see epistemology as necessary.)

So, it's not a simple thing. It's not necessarily complex, either. But it is compound! I'm being deliberately philosophical, here, so bear with me. Bacon thought that science should be done by many in a huge enterprise of creating knowledge (the "great instauration") . He didn't want to see it the province of a few geniuses. He wanted a huge labor force of ordinary, mere mortals. So his view of induction is that it is a simple thing. But many inductions have to be performed to assemble serious knowledge from it. That makes it "compounded" from all the work performed.

How can one improve one's inductive powers? I have no magic bullet to offer. There is no Royal Road to knowledge. I guess that you can learn a few pointers from reading Bacon's Nuovum Organum, and followers. You can learn other techniques and more modern refinements, such as looking for a dose-response relationship to prove causation of the effect, rather than simply looking for presence/absence. You can learn the principles of double-blind testing, for the elimination of observational bias. But the bottom line is that serious theories are based upon a huge number of inductions. It's not just one induction that leads to one theory.

Rigor means “disciplined”, but not in the sense that discipline means “punishment”, instead, it’s in the way that “disciplined” means “systematic behavior”.

As much as we move forward in mathematics by adding more deductive proofs, those deductive proofs must be backed up by more proofs, that must be backed up by more proofs, which must eventually be backed up by our axioms, and our axioms must be proven to be consistent.

Full rigor, in mathematics, is the systematic application of proofs, leaving little to nothing to the intuition, leaving nothing undefined, and leaving no uncertainty.

That bein

Rigor means “disciplined”, but not in the sense that discipline means “punishment”, instead, it’s in the way that “disciplined” means “systematic behavior”.

As much as we move forward in mathematics by adding more deductive proofs, those deductive proofs must be backed up by more proofs, that must be backed up by more proofs, which must eventually be backed up by our axioms, and our axioms must be proven to be consistent.

Full rigor, in mathematics, is the systematic application of proofs, leaving little to nothing to the intuition, leaving nothing undefined, and leaving no uncertainty.

That being said, not everything in mathematics uses “full rigor” as there are things at the foundational level still being questioned*. There are also very few things that are known to extreme probabilities, but not proven for every case. However, even it’s still full rigor when we are up front and descriptive about that which we do know vs that which we still don’t know. Therefore, full rigor is also about using the right language to eliminate ambiguity

*(but trust me, the level of questioning in the foundations is so deep and beyond numbers and early studied objects, explaining things like ‘resolving parametricity and canonicity via the addition of transpension and fresh weakening types in order to provide a framework to represent equality and its properties between different categories of objects’ really starts to question how we represent objects in our minds, as well as on computers)

First, let me address the elephant in the room. Can you improve your aptitude and reasoning skills? Yes, but only till certain extent. Do not expect your way out towards uncovering the mysteries of universe and existence. What you can expect is that you will be able to perform certain tasks that require mental effort easily.

So, now coming to your question: the how?

If you are preparing for some exam: then, I will shortly say, just practice more with an eye towards understand the concepts.

If you are looking for improving this in general sense: then, you need to explore various avenues of good th

First, let me address the elephant in the room. Can you improve your aptitude and reasoning skills? Yes, but only till certain extent. Do not expect your way out towards uncovering the mysteries of universe and existence. What you can expect is that you will be able to perform certain tasks that require mental effort easily.

So, now coming to your question: the how?

If you are preparing for some exam: then, I will shortly say, just practice more with an eye towards understand the concepts.

If you are looking for improving this in general sense: then, you need to explore various avenues of good thinking. This encompasses critical thinking, logical thinking, creative thinking and also construction of a foundation based no which you will make your mental models to understand the world, etc.

So, now you know, what it means to really improve the reasoning skills and aptitude. These are the following things you need to do:

- Critical thinking: understand what is an argument. How the argument is structured. What are the implications of this argument. In what context does this argument fit into.
- Logical thinking: Pick up any book written by Raymond Smullyan.
- Search on Quora for improving the above two. There are good answers.

Working on only developing these two aspects will significantly improve your processing i.e. you will start tilting towards good thinking.

Let me start by sharing this article from “The Critical Thinking Co.”:

Guide To Inductive & Deductive Reasoning

They provide a nice framing for inductive reasoning:

“Inductive reasoning moves from specific details and observations (typically of nature) to the more general underlying principles or process that explains them (e.g., Newton's Law of Gravity). It is open-ended and exploratory, especially at the beginning. The premises of an inductive argument are believed to support the conclusion, but do not ensure it. Thus, the conclusion of an induction is regarded as a hypothesis. In the Inductive

Let me start by sharing this article from “The Critical Thinking Co.”:

Guide To Inductive & Deductive Reasoning

They provide a nice framing for inductive reasoning:

“Inductive reasoning moves from specific details and observations (typically of nature) to the more general underlying principles or process that explains them (e.g., Newton's Law of Gravity). It is open-ended and exploratory, especially at the beginning. The premises of an inductive argument are believed to support the conclusion, but do not ensure it. Thus, the conclusion of an induction is regarded as a hypothesis. In the Inductive method, also called the scientific method, observation of nature is the authority.”

One way to improve your inductive reasoning skills is through practice. Searching online for “inductive reasoning tests” provides lost of possibilities. For example:

https://www.wikijob.co.uk/content/aptitude-tests/test-types/inductive-reasoning-test

Taking these tests can help highlight areas where you already have a foundational level and where there are opportunities to improve.

Here is some guidance on how to succeed at inductive reasoning tests:

Tips to Succeed at Inductive Reasoning

Hope these resources sparks some thinking for how you can go about improving your inductive reasoning skills.

i could add double digit numbers in my head when i was three or four. Then i had math teachers that were total failures until I took Algebra, so i’d say Algebra. Back then they had these ridiculous notions that kids were mature enough to learn the number systems at a young age back then (1960’s) was foolishness.

I actually had a math teacher that said if you subtract 5 from 4 the answer is zero. I knew it wasn’t zero because if my mom spent more money than was in the checkbook it was a non-zero number that made my dad unhappy!

The start of mathematics reasoning is when the student is asked figur

i could add double digit numbers in my head when i was three or four. Then i had math teachers that were total failures until I took Algebra, so i’d say Algebra. Back then they had these ridiculous notions that kids were mature enough to learn the number systems at a young age back then (1960’s) was foolishness.

I actually had a math teacher that said if you subtract 5 from 4 the answer is zero. I knew it wasn’t zero because if my mom spent more money than was in the checkbook it was a non-zero number that made my dad unhappy!

The start of mathematics reasoning is when the student is asked figure things out and ask questions. This doesn’t get better in Algebra. We given the “quadratic formula” OK what happens when you get to quintic equation? is the world still filled with mathematics benevolence? NO! It’s unsolvable!

The you get to Geometry and the Tri-geometry. Look kids you can prove all these things just like the Greeks did kids! And these angles can be expressed with niche number pie (but i thought you said that was irrational) Wow! how about others ? They are unsolvable!

Years ago a found an out print book called “Mathematics Made Easy” which I totally loved! It is both good book on foundations and totally hilarious! It starts with complicated and simple may depend on your point of view and goes from there. My mother said “stop read that book (aloud)!”

So higher learning in math may start with a good analytical smack upside the head. Trying to make it easy was a cruel joke! You may think I’m insane, but i ask is that a problem? Really, for whom?

Read, read and read more. And argue. And debate. And read more.

Logic is not a definitive thing that can be "achieved" or "acquired" like a degree. It is a continuously evolving, never ending process that gets honed the more you use it. Humanity and Logic have always changed over the past, and despite people claiming to have a fixed idea of Logic being unchangeable - It is very abstract. Reading different perceptions and perspectives of others on logic will help you form your own version of it.

Take for example, It was illogical for us to assume the earth could ever be spherical- since if we c

Read, read and read more. And argue. And debate. And read more.

Logic is not a definitive thing that can be "achieved" or "acquired" like a degree. It is a continuously evolving, never ending process that gets honed the more you use it. Humanity and Logic have always changed over the past, and despite people claiming to have a fixed idea of Logic being unchangeable - It is very abstract. Reading different perceptions and perspectives of others on logic will help you form your own version of it.

Take for example, It was illogical for us to assume the earth could ever be spherical- since if we carried out an experiment way back then- and poured water over a small sphere, the water would simply fall off. Since the seas never "fell off", logically the earth could be anything but spherical. The concept of Gravity quite changed that Logic forever.

For reasoning:

Start with building your concept of each topics.

1. Learn number of each letter eg. A-1 ,B-2, C-3,D-4 etc. Practice it by writing random Letters and answer them. It will complete your analogy part.

2. For series type question: Try each pattern like difference between two numbers could be addition , multiplication, square of number, cube of number etc. Practice no. Of questions then you will know various logics in numbers.

3. No. Of triangle: you can watch any teacher videos on YouTube. Then practice by book.

4. Direction type question: learn direction sense and move according to dire

For reasoning:

Start with building your concept of each topics.

1. Learn number of each letter eg. A-1 ,B-2, C-3,D-4 etc. Practice it by writing random Letters and answer them. It will complete your analogy part.

2. For series type question: Try each pattern like difference between two numbers could be addition , multiplication, square of number, cube of number etc. Practice no. Of questions then you will know various logics in numbers.

3. No. Of triangle: you can watch any teacher videos on YouTube. Then practice by book.

4. Direction type question: learn direction sense and move according to direction asked in question.

5. Puzzles: learn concept of each puzzle individually from youtube or teacher and then practice hard.

For mathematics

- Calculations: start learning tables, cube ,square square root, cube root,BODMAS and then apply into questions.
- Concept: Do not go directly to question first of all start with concept of that particular chapter.
- Type of questions: Analysis the type of questions asked in exam and focus more on that types of questions.
- Revision: Revision is the key of success, revise all previous concepts formulas, calculations part regularly. So that yoy will not forget at the time of examination.

Lastly practice. Practice. Practice. As much you do your skill will improve.

Thank you.

Hello,

If by “getting” you mean succeeding in and understanding the material of a mathematics degree, I would say this is true.

Mathematicians require logic and reasoning to solve problems they encounter - it’s a skill that will develop throughout the years (for me, anyway).

I have found that problems I struggled with, maybe in my first year of my degree, I can attempt and obtain a correct solution to now, because I am really thinking about the question and using reason to work my way through it.

Hope this helps,

Natasha